Multiperiod portfolio optimization for asset–liability management with quadratic transaction costs

If you believe this document infringes copyright then please contact the KAR admin team with the take-down information provided at http://kar.kent.ac.uk/contact.html 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.

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. 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.
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.

Problem formulation
Consider a capital market with 1  n 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  which is supposed to be positive definite. The investor allocates his initial wealth 0 W 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 , n i ,..., 2 , 1  , as the investment amount on risky asset i which is allowed short selling and } ,..., , as the adjustment amount on risky asset i at period t . Therefore, the investment amount on riskless asset is , and the adjustment amount on riskless asset based on the self-financing assumption.
In addition, suppose that the investor has an exogenous liability. The initial liability is 0 L . Let t q be the return of liability at the t -th investment period, where ) , (   t t q e is statistically independent. We diagonalize the co-variance vector about the liability and risky assets, denoted as )) ,  5 and the surplus at period t can be expressed as We follow the quadratic transaction costs adopted in Gârleanu and Pedersen (2013).
Under this setting, the transaction cost (TC) associated with trading volumes where  is a symmetric positive-definite matrix measuring the level of total trading costs.
Note that the transaction cost t C depicts the expense arising from changes on investment amounts at period t rather than trading shares shown in Gârleanu be the strategy at period t , and then the multiperiod asset-liability management problem with quadratic transaction costs can be expressed as: 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 6 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.
is the optimal strategy for the auxiliary is also the optimal strategy for the problem denotes the expectation of the investors' final surplus when he invests according to the optimal strategy According to Theorem 2.1, the pre-commitment strategy can be obtained by the following steps: 1) We first construct the auxiliary problem which is a separable structure in the sense of dynamic programming.
2) Through the idea of dynamic programming, we obtain the solution of the auxiliary problem, and is a function of  .
3) By iterating each period of the strategy with the state transition equation of surplus, it is easy to find the expected final surplus , which is a function of  . Then, by using equation 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.
Then ṽ  is call as the time-consistent strategy if for all be the time-consistent strategy at period t , Definition 2.1 makes it possible to solve the problem by the following procedures: 2) Given that the decision maker is the optimal strategy by optimizing objective function )) , 3) Generally, given that the forthcoming decision makers For a mean-variance investor, the pre-commitment as well as time-consistent strategies are available. We will show them in the following sections.

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: is the cost-aversion coefficient. For a specific investor,  and  are constant.

Pre-commitment strategy for problem
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 )) , , For convenience, we list the notations of this section as following. Define: The following notations are defined for 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 )) , , can be given in the following Theorem 3.1.
and optimal value function And then, in accordance with Theorem 2.1, we can obtain 3 0 (3.5) The optimal investment strategy of problem )) , ( (   P , the corresponding expectation and variance of surplus and expected transaction cost for 1 ,..., 2 , respectively, as follows (3.9) Remark 3.1. When the investor has no liability, that is and the expectation and variance of surplus and the expected transaction costs for (3.13)

Remark 3.2.
When ignoring the transaction cost, that is , the pre-commitment strategy can be acquired by setting 0   in equations (3.10). In addition, if the liability ) 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.

Time-consistent strategy for problem
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.
By applying Bellman's principle of optimality, the time-consistent investment strategy of is given in the following theorem. (3.14) and the expectation of surplus is (3.17) Proof. See Appendix C.

Remark 3.3. If the investor have no liability, that is
and the expectation and variance of surplus and the expected transaction costs, respectively, (3.21) 14 Furthermore, the Theorem 3.2 still generalizes the situation without transaction cost when 0   , and the situation without liability and transaction cost.  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: 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.

Pre-commitment strategy for problem
The analytical solution and the optimal value of objective function to problem )) , , ( (    A are derived by dynamic programming approach.
Define: T  T  T  T  T   R  Q  I  P  I  N  I  I  M  ,  ,  2 , , The following notations are defined for where I is the n-dimensional column vector of element 1, and  is a unit matrix.

Theorem 4.1. The optimal strategy
And the expectation and variance of surplus and the expected transaction costs, respectively, And the expressions of expectation and variance of surplus and the expected transaction costs (4.10) This implies that the Theorem 4.1 can generalize the situation without liability. 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. Define:

Time-consistent strategy for problem
Subsequently, by applying the procedures of time consistent strategy, we have the following conclusions.
, the time-consistent investment strategy of problem )) , the corresponding expectation and variance of surplus and the expected transaction costs, respectively, is： (4.14)

Remark 4.3. Similarly, if the investor does not have any liability, that is
, then the time-consistent strategy reduces to (4.15) and the expectation and variance of surplus and the expected transaction costs, respectively, Thus the same to Theorem 3.2, the Theorem 4.2 generalizes three situations as well.

Numerical simulations
In this section, some numerical simulations are given, which provide twofold According to the conclusions shown in previous sections, we can find that

Example 5.1 Comparison of strategies with/without cost
Although the empirical evidence shows that the transaction cost affects the strategy, it fails to quantify the extent of the change intuitively. Thus, we compare the strategies of considering the transaction cost and that of ignoring it.

Example 5.2 Comparison of the frontiers under different strategies
In order to better understand the difference among different investment strategies, we will discuss the frontiers under the following two situations: When other parameters remain unchanged, different cost aversion coefficients will produce different frontiers. The detailed simulation results are shown in Fig 5.2.  Fig.5.2, we can draw two conclusions. One is, for the given risk level, the expected net surplus of pre-commitment strategy is better than that of time-consistent strategy no matter that there is a riskless asset or not in the asset pool. In other words, we can obtain higher income by following the pre-commitment strategy. This can be explained by that the pre-commitment strategy is the global optimal investment strategy for the initial period, while the time-consistent strategy only considers local incentives and ignores global objectives. The existence of the quadratic transaction cost does not affect the superiority of the pre-commitment strategy. The other interesting conclusion is that when the value of  is particularly large, the gap between the frontier of pre-commitment strategy and that of time-consistent strategy have been reduced. Comparatively speaking, the cost constraint is more punitive to the pre-commitment strategy. If the investor adopts the pre-commitment strategy without considering the transaction cost, then it will lead to ineffective investment strategy, especially for the individual investor with higher cost aversion.

Example 5.3 Impact of cost-aversion coefficient on different frontiers
To explore the impact of cost-aversion coefficient on frontiers, we set  is, in turn, equivalent to 0,0.8,1.6 and 2.4. Fig. 5.3 shows the sensitivity of the corresponding frontier 23 under different strategies to the cost coefficients. More importantly, no matter how large the cost aversion coefficient is, the produced cost is relatively small for the time-consistent strategy.

Example 5.4 Impact of parameter  on different frontiers
The positive definite matrix  in the quadratic transaction cost function can be diagonalized into a matrix consisting of eigenvalues, which dominate the corresponding unit cost of risky assets. In this example, we will discuss the impact of these eigenvalues on different frontiers.
Here, we set the matrix  as Table 5.1 and  equals to 0.5. Fig. 5.4 shows the frontiers of different strategies when  takes different value.

Conclusion
This paper provides the highly tractable multiperiod asset-liability management frameworks for the study of optimal trading strategies in presence of quadratic transaction costs. For different investment setting (with/without riskless assets), the pre-commitment and for given  and  . Similarly, for problem is the set of the optional solution for given  ,  and  . And denote (1) We firstly proof that for any Consider the function Apparently, it is conflict with the assumption On the other hand, due to And because of then the first order necessary condition of optimal solution about * is also the optimal control of the problem

Appendix B
Proof of Theorem 3. 1 We adopt the dynamic programming of reverse solving method to solve the problem )) , , , applying dynamic programming principles gives rise to  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  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 Applying the first order condition about 1   T v yields the following optimal strategy  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 Next, for every 2 ,..., 2 , , by using mathematical induction we can suppose .' According to the state transition equations, there is Applying the first order condition about t v  yields the following optimal strategy Substituting (A.7) into (A.6) and simplifying the resulting equation yields . '  T  T  T  T  T   T  T  T  T  T  T  T  T  T   T  T  T  T  T  T  T  T Applying the first order condition about 1 -T v  yields the following optimal strategy (B.2) And the condition expectation and variance of final wealth and condition expected cost, respectively, is  ' ) ( ) )( ( ) (  T  T  T  T  T  T   T  T  T  T  T  T  T   v  z  L  y  v  x   L  q  E  v  v  e  E  S  E   (B.3)   1  1  1  1  1  1  1  1  2  1  1  1  1  1   1  1  1  0  1  2  1  1  1  1  1  1  1 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  T  T  T  T  T  T  T  T  T  T  T  T It is easy to find that Theorem 3.2 holds for period 1  T .
Assume that Theorem 3.2 also holds for 1  t , then for t , we can obtain that Applying the first order condition about t v  yields the following optimal strategy And the condition expectation and variance of final wealth and condition expected cost for period t , respectively, is (B.10) It is easy to find that Theorem 3.2 also holds at period t for 1 ,..., 1 , 0   T t . By mathematical induction, we complete the proof of Theorem 3.2. Q.E.D