Abstract
The assetliability management problem with cash flow under an uncertain exit time has been investigated in this article, which is based on the fundamental framework of the meanvariance model in the multiperiod version. The liability and random cash flow will affect asset optimization, while the investor may be forced to withdraw from investments with a random probability at each period in our model. The closedform expressions for the meanvariance optimal portfolio selection and its corresponding efficient frontier are obtained by employing the meanfield formulation and dynamic programming approach. Moreover, some numerical examples are provided to illustrate the validity and accuracy of the theoretical results.
1 Introduction
With an explosive development of the economy in the recent 50 years, it is becoming more common that lots of private assets have been invested in the financial market. After a series of financial crises, the significance of handling private assets has been attached considerably. Meanvariance formulation is a famous tool that aims at balancing the risk and return of the investment. Owing to the seminal work of Markowitz [1], the meanvariance model has provided a fundamental basis for designing the optimal strategy balancing the contradiction between return and risk. Hundreds of applications and extensions have been developed over the past decade. For instance, Merton [2] derived the analytical expression of the meanvariance efficient frontier in a singleperiod setting. Li and Ng [3] developed the meanvariance model from the single period to the dynamic discretetime version and derived the analytical solution by using the embedding method to overcome the difficulty of nonseparability. Zhou and Li [4] used the same technique and further introduced the stochastic linear quadratic control as a general framework to solve the continuoustime meanvariance portfolio selection problem. Li et al. [5] developed it with the noshorting constraint. Moreover, some recent approaches [6,7] based on enhanced index tracking are employed to deal with portfolio optimization.
There is no doubt that the embedding method is indeed a classic way to solve the problems with the nonseparable property. We also need to admit that this method is liable to lead to complicated calculation and inefficiency during the derivation of the optimal portfolio selection if the problem has other constraints, such as uncertain exit time, assetliability management, and serial correlated returns or risk control over bankruptcy. Typically, we will prefer another method called meanfield formulation to the embedding scheme, where a long list of notations should be established, and an auxiliary problem should be assumed.
The meanfield formulation is a simple but powerful tool to derive the optimal strategy of a multiperiod meanvariance portfolio selection problem. By using this method, which was first introduced by Cui et al. [8], we can resolve the meanvariance problem with many other additional constraints and derive the optimal strategy in a simpler and more direct manner. Yi et al. [9] used the meanfield method to study the meanvariance model under the uncertain exit time condition but did not consider the cash flow and liability. Cui et al. [10] extend it to the assetliability management, but none of them considered the situation of random cash flow or investigated the closedform of computational formulas for a series of coefficients.
Yao et al. [11] studied the meanvariance model with a given level of expected terminal surplus. Li and Xie [12] studied the optimal investment with stochastic income under the uncertain exit time. They derived the analytical optimal strategy and explicit expression of the efficient frontier by using the Lagrange method and traditional dynamic programming with the additional conditions of endogenous liabilities if the investors exit the market randomly. Wu and Li [13] investigated the multiperiod meanvariance model with different market states and stochastic cash flows. A reinforcement learning framework is employed to investigate the continuoustime meanvariance portfolio selection [14]. Ni et al. [15] derived equilibrium solutions of multiperiod meanvariance and established a general theory to characterize the openloop equilibrium control problem. However, all of the literature did not consider the correlation among cash flow, asset, and liability, which should be taken into account in the real world because the random cash flow would be affected by the return rate of companies. For example, the government will provide funding to companies in terms of their past performance. Furthermore, since the analytical solution of the meanvariance model contains the correlation coefficient, the optimal strategy will be changed due to different return rates among the asset, cash flow, and liability. Moreover, the uncorrelated case can be regarded as a special case of the correlated one, of which the correlation coefficient is zero. We extended the special case to the general case.
In this paper, we employed the meanfield formulation [8,9,10] to study the general case of correlation in which the financial parameters are correlated at every period. On the basis of the aforementioned meanfield formulation, we have added some additional conditions such as random cash flow and liability to improve the accuracy of the investment strategy. During each time period, the cash flow and risky investment returns are random variables, while the riskfree investment return is deterministic. Furthermore, we have derived the analytical solutions of the meanvariance model which is lacked by using the embedding method [3,11, 12,13]. Employing the embedding method, the classical model mentioned earlier has certain limitations since they need to define a deterministic expectation of surplus, which is a singleobjective optimization problem. Besides, the numerical solution needs some algorithms to compute the corresponding best auxiliary parameter or Lagrangian parameter, which will bring the inaccuracy and complexity in simulation. However, the meanfield formulation is more clear and powerful, which offers an analytical solution scheme in solving the nonseparable problems as the principle of optimality no longer applies. When both cash flow and meanfield formulation are presented in the same model, we shed light on the explicit solutions of the optimal portfolio under meanvariance criteria. In this paper, we are not only concerned about the return rate but also concerned with the volatility in the objective function in terms of a multiobjective optimization problem. We study the portfolio selection problem by adopting the mean field, and consider the cash flow, liability, etc. base on the mean variance model. Compared with the numerical solution, the analytical solution we derived in this paper is more efficient and applicable when the aforementioned additional conditions are added to our model.
The rest of the paper is structured as follows. We construct a meanvariance portfolio selection problem with cash flow and define the meaning of some symbols in Section 2. In Section 3, the considered model is equivalently transferred into a linear quadratic optimal stochastic control problem in the meanfield type. Then, we identify the optimal portfolio strategy with closedform expressions by adopting the dynamic programming approach in Section 4. Some numerical examples are provided in Section 5 to illustrate the accuracy and efficiency of the optimal strategy. Finally, the conclusion and future work are given in Section 6.
2 Multi period meanvariance portfolio selection model
We assume the financial market has one liability, one riskfree asset, and
In different time periods
Then, for
Let
where
Therefore,
At the beginning of every period
The investor plans to optimize the portfolio selection during the whole time period. However, the investment might be forced to be changed or abandoned at an uncertain time
The main investigation of this model is to find the optimal portfolio selection,
where
Then, we can rewrite the aforementioned model as follows:
Since the smoothing property is no longer valid on the variance term, we cannot decompose the nonseparable problem into a stage wise backward recursion formulation, which can be tackled with traditional dynamic programming method. We solve it by employing the meanfield method.
3 Meanfield formulation
First, we construct the meanfield type of model (3). According to the independence between
with
Combining the dynamic equations in (3) and (4), we have
Therefore, we can equivalently reformulate problem (3) into a linear quadratic optimal problem in the meanfield type.
Thus, we are able to solve it by the dynamic programming method since it is separable.
4 The optimal strategy
With the notations given in (1), the seven parameters of the sequence
with boundary conditions defined as follows:
The solution scheme adopted in this paper involves two steps. The first step is to construct the costtogo functional and derive the backward recursion. The second step is to prove that it still holds at each period according to mathematical induction. Thus, the optimal portfolio strategy can be obtained in the following theorem.
Theorem 1
Assume that the return rates among asset, liability, and cash flow are correlated. Thus, we have the optimal portfolio selection of problem (6) as follows:
The expected value of optimal wealth can be derived as follows:
for
If the additional condition of liability is not considered in our case, the original model (6) would be degenerated to the one mentioned by Yao et al. [11], which will be introduced in the following corollary.
Remark 1
Assume that an investor participates in the initial investment under uncertain exit time without liability. Thus, the degenerated problem is equivalently reformulated as the following meanvariance model.
The optimal strategies of problem 7 are represented as follows:
Thus, we get the optimal expected level of wealth
The optimal strategy of the model in Corollary 1 can be obtained according to Theorem 1, which is consistent with the results derived by Yao et al. [11]. Therefore, the accuracy of the solution derived in this paper has been verified. In comparison, Zhu et al. [16] analyzed the Lagrangian problem via the embedding method and were unable to obtain an analytical form of the optimal objective value function. Thus, they invoked a primedual iterative algorithm to identify the optimal Lagrangian multiplier vector. Moreover, compared with the classical embedding method, which needs a Bellman equation and the Lagrangian multiplier, the meanfield formulation has been employed in this paper, which avoids the complicated computation. In the following section, a few numerical examples from realworld applications are given to demonstrate the efficiency of the obtained optimal strategy.
5 Numerical example
According to the data given in the study by Elton et al. [17], we investigate a portfolio selection consisting of S&P 500 (SP), the index of emerging market (EM), and small stock (MS) of the U.S. market. Moreover, we consider uncertain exit time and cash flow in the model. Table 1 presents three different assets, a liability, and a random cash flow, and it also presents the expected values, variances, and the correlation coefficients among them. The annual risk free return rate is set as
SP  EM  MS  Cashflow  Liability  

Expected return 



1 

Standard deviation 





Correlation coefficient  
SP  1  0.64  0.79 


EM  0.64  1  0.75 


MS  0.79  0.75  1 


Cashflow 



1 

Liability 




1 
Thus, for every period
The correlation coefficient between cash flow and
and
In addition, we define the correlation of the cash flow and liability
Then, we have
Assume that
Substituting the data in the equations, we have
Example 1
An example with the terminal exit time
The probability mass function
for
Under the certain exit time, we derive the final optimal surplus as follows,
Example 2
An example without liability under uncertain exit time
Consider the example as corollary. Here, we ignore the information of liability, i.e., ignore the last line and last column of Table 1 and do not fix the terminal expectation but balance the variance and expectation by the tradeoff parameter.
Assume that an investor plans a fiveperiod investment with an initial wealth
To investigate the impact of uncertain exit time on the optimal policy and efficient frontier clearly, we choose four different probability mass functions at the exit time
where
Then, the optimal expected wealth level
which are given by
Therefore, the optimal strategy is specified as follows:
where
which are given as follows:
Thus,we have
Figure 1 depicts the efficient frontier with different probability mass function of the exit time. We can see that the one exits at the terminal time gets the most expected wealth return at the same risk level compared with others. It is also indicated that if the investment is more stable, the investors can obtain higher expected returns at the same level of the risk, which is consistent with the real life.
Example 3
An example under uncertain exit time with liability
The probability mass function of an exit time
Thus, the optimal expected value of assets in different time periods is given by
Suppose the initial wealth of the investor
Furthermore, the final value of mean and variance under the optimal strategy are
Following Example 2, we choose four different probability mass functions at the exit time
Figure 2 is the efficient frontier of M–V model with liability and random cashflow under uncertain exit time. It can be seen that as the expectation go up, the more stable the investment, the less risk it takes, which has the same conclusion as Figure 1. Actually, Example 2 is a special case of Example 3, where we degenerate the term of liabilities to zero.
6 Conclusion
The focus of the paper is placed on investigating the optimal strategy of multiperiod meanvariance model with cash flow, and liability under uncertain exit time. It is a nonseparable dynamic programming problem that cannot be solved by the traditional method. In this paper, we transform the original model into a meanfield type and apply a dynamic programming approach and matrix theory to derive the optimal strategy explicitly. Our methods are shown to be much more efficient and accurate compared with other methods in the literature. For further research, we will try to employ the meanfield method to derive the meanvariance model with various additional conditions such as regime switching, bankruptcy constraints, and time inconsistency.
Acknowledgements
The authors express their appreciation for the anonymous referees comments and suggestions.

Funding information: This work was sponsored by the Philosophy and Social Science Planning Project of Guangdong Province (Grant No. GD20YGL12), Basic and Applied Basic Project of Guangzhou City (Grant No. 202102020629), Philosophy and Social Science Planning Project of Guangzhou City (Grant No. 2021GZGJ48), National Natural Science Foundation of China (Grant No. 71771058), and Guangdong Basic and Applied Basic Research Foundation (Grant No. 2020A1515110991).

Conflict of interest: The authors state no conflict of interest.
Appendix
The Proof of Theorem 1
Proof
Given an information set
From time
Next, we will prove that the aforementioned formulation (7) still hold at time