A mean-field formulation for multi-period asset-liability mean-variance portfolio selection with probability constraints

This paper is concerned with studying an optimal multi-period asset-liability mean-variance portfolio selection with probability constraints using mean-field formulation without embedding technique. We strictly derive its analytical optimal strategy and efficient frontier. Numerical examples shed light on efficiency and accuracy of our method when dealing with this class of multi-period non-separable mean-variance portfolio selection problems.


1.
Introduction. Mean-variance portfolio selection refers to the design of optimal portfolios balancing gain with risk, which are in expressions of expectation and variance of the terminal return, respectively. After Markowitz's [14] seminal work for a single-period, research on mean-variance portfolio selection problems have been well developed. For instance, by an embedding technique, Li-Ng [11] extended Markowitz's model to a multi-period setting and derived analytical optimal portfolio and efficient frontier; Zhang-Li [23] generalized their work to the case with uncertain time horizon when returns are serially correlated. Zhou-Li [24] studied continuous-time mean-variance problem using the similar technique and obtained closed forms of optimal portfolio and efficient frontier; Li-Xie [13] gave a generalization of their problem in several aspects. Fu-Lavassani-Li [7] and Li-Zhou-Lim [12] investigated dynamic mean-variance portfolio selection with borrowing constraint and no-shorting one, respectively. Cui-Gao-Li-Li [4] tackled multi-period problems. Finally they searched for the Lagrangian multiplier using primal-dual method. The construction of auxiliary problems and the inexplicit optimal objective function may possibly involve some unnecessary and complicated expressions or computational errors, resulting in complicated or even inaccurate formulas. We employ mean-filed formulation to successfully reformulate the nonseparable problem to a mean-field linear-quadratic stochastic control problem which can be solvable by the classical dynamic programming approach, and strictly derive analytical optimal strategy of this problem and its efficient frontier. Compared with Li-Li [10], our results have much simpler formulas.
The rest of the paper is organized as follows. In section 2, we present the meanfield formulation for the multi-period asset-liability mean-variance portfolio selection with probability constraints. We strictly derive the optimal strategy and the corresponding efficient frontier in section 3. Numerical examples are presented in section 4 to show the efficiency of the mean-field formulation to solve the nonseparability of multi-period mean-variance portfolio selection problem with probability constraints.
2. Formulation. Assume that the capital market consists of one risk-free asset, n risky assets and one liability. An investor joining the market at the beginning of period 0 with an initial wealth x 0 and initial liability l 0 , plans to invest his/her wealth within a time horizon T . He/she can reallocates his/her portfolio at the beginning of each following T − 1 consecutive periods. At time period t, the given deterministic return of the risk-free asset, the random returns of the n risky assets, and the random return of the liability are denoted by s t (> 1), vector e t = [e 1 t , · · · , e n t ] and q t , respectively. The random vector e t = [e 1 t , · · · , e n t ] and the random variable q t are defined over the probability space (Ω, F, P ) and are supposed to be statistically independent at different time period. We further assume that the only information known about e t and q t are their first two unconditional moments, t ], · · · , E[e n t ] , E[q t ] and (n + 1) × (n + 1) positive definite covariance Let x t and l t be the wealth and liability of the investor at the beginning of period t respectively, then x t − l t is the surplus. At period t, if π i t , i = 1, 2, · · · , n is the amount invested in the i-th risky asset, then, x t − n i=1 π i t is the amount invested in the risk-free asset. We assume in this paper that the liability is exogenous, which means it is uncontrollable and cannot be affected by the investor's strategies. Denote the information set at the beginning of period t, t = 1, 2, · · · , T − 1, as F t = σ(P 0 , P 1 , · · · , P t−1 , q 0 , q 1 , · · · , q t−1 ) and the trivial σ-algebra over Ω as F 0 , where P t = (P 1 t , · · · , P n t ) = (e 1 t − s t , · · · , e n t − s t ) is the excess return vector of risky assets. Therefore, E[·|F 0 ] is just the unconditional expectation E[·]. We confine all admissible investment strategies to be F t -adapted Markov controls, i.e., π t = (π 1 t , π 2 t , · · · , π n t ) ∈ F t . Then, P t and π t are independent, {x t , l t } is an adapted Markovian process and F t = σ(x t , l t ).
The mean-variance model for multi-period asset and liability portfolio selection with probability constraints is to seek the best strategy, π * t = [(π 1 t ) * , (π 2 t ) * , · · · , (π n t ) * ] , t = 0, 1, · · · , T −1, which is the optimizer of the following stochastic optimal control problem, where w > 0 is the trade-off parameter between the mean and the variance, a t is the probability of the wealth less than the liability at period t. Since the probabilistic constraint Pr(x t ≤ l t ) is not easy to conquer in dynamic portfolio selection, we turn it to its upper bound Var(x t − l t )/(E[x t − l t ]) 2 by Tchebycheff inequality. Then the mean-variance model (1) can be equivalently re-written to the following problem, The optimal solution to problem (2) is feasible in problem (1), thus serving as an approximated solution to problem(1). To solve problem (2), we consider the following Lagrangian minimum problem, l t+1 = q t l t , t = 1, · · · , T − 1, is the vector of Lagrangian multipliers. Due to the variance operation does not satisfy the smoothing property, problem (3) is nonseparable in the sense of dynamic programming, i.e., it can not be decomposed by a stage-wise backward recursion and then is difficult to solve directly. We tackle it by mean-field formulation. For t = 0, 1, · · · , T − 1, taking the expectation operator of the dynamic system specified in (3) we can drive since P t and π t , q t and l t are independent. Combining the dynamic system of (3) and (4) yields the following for t = 0, 1, · · · , T − 1, Then the state space (x t , l t ) and the control space , respectively. Although we can select the control vector E[π t ] and π t − E[π t ] independently at time t, they should be chosen such that Problem (3) can be now reformulated as the following mean-field type of linear quadratic optimal stochastic control problem (4), } satisfies dynamic equation (5), It is indeed a separable linear quadratic optimal stochastic control problem which can be solved by classic dynamic programming approach.
3. The optimal strategy. Before deriving the main results, we present some useful lemmas.

Lemma 3.1 (Sherman-Morrison formula).
Suppose that A is an invertible square matrix and µ and ν are two given vectors. If then the following holds, Proof. (i) Applying Sherman-Morrison formula yields (ii) Applying Sherman-Morrison formula yields (iii) Applying the above (ii) yields (iv) Applying the above (ii) yields Assume that the returns of assets and liability are correlated at every period. For simplicity, we define the following backward recursions for eight deterministic sequences of parameters, for t = T − 1, T − 2, · · · , 0, with terminal conditions Remark 1. When the returns of assets and liability are uncorrelated, which is to say,

XIANPING WU, XUN LI AND ZHONGFEI LI
And others are the same as the correlated case.
Theorem 3.3. Assume that the returns of assets and liability are correlated at every period. Then, the optimal strategy of problem (3) is given by for t = 0, 1, · · · , T − 1.
Proof. We prove the main results by dynamic programming approach. For the information set F t , the cost-to-go functional at period t is computed by The cost-to-go functional at terminal time T is Assume that the cost-to-go functional at time t + 1 is the following expression We prove that the above statement still holds at time t. For given information set F t , i.e., knowing Since any admissible strategy of (E We first identify optimal (E[π * t ], π * t − E[π * t ]) by minimizing the following equivalent cost functional,

XIANPING WU, XUN LI AND ZHONGFEI LI
It is easy to see that π * t −E[π * t ] can be expressed by the linear form of states and their expected states, and E[π * t ] can be constructed by the linear form of the expected states, i.e., In order to get the explicit expression of the cost-to-go functional at time t, we substitute π * t − E[π * t ] and E[π * t ] back and derive Also, which implies Hence, combining with (9) and (10), we derive the desired result (7). Finally, we show that this optimal strategy satisfies the linear constraints. t − E[π * t ]] = 0 holds for all t. We can simply reformulate the optimal strategy (7) to the following form: wherē It is obviously that the derived analytical optimal portfolio policy consists of three terms: the investor's current wealth, current liability and risk attitude specified by w and a t , which are also a function of the initial wealth x 0 and the initial liability l 0 . In other words, it is of a feedback form, but not Markovian. At each period t, the optimal control policy depends on information from the given information set including the current state (x t , l t ) and the initial state (x 0 , l 0 ).

Remark 2.
When the returns of assets and liability are not correlated, Based on the proof of Theorem 3.3, the optimal objective of problem (3) is as follows, In fact, J 0 (·) is convex in λ. From its explicit form, we can find optimal Lagrangian multiplier vector λ * simply by steepest descent algorithm or interior point algorithm directly in the code via MATLAB. Compared with Li-Li [10] using the embedding scheme, they could not obtain the optimal objective value function J 0 (·) analytically. Then they proposed the prime-dual iterative algorithm to search for the optimal Lagrangian multiplier vector λ * : First, for a given ω, a system of linear equations were solved to get the embedding parameter vector λ(ω), then the optimal policy of Lagrangian problem and a feasible decent direction were computed. Finally, a line search along the feasible decent direction was carried out to determine the optimal step-size. The inexplicit optimal objective function might possibly involve some computational errors, resulting in even inaccurate results. By the meanfield formulation, we can derive directly the analytical optimal policy. It is an efficient way to solve the asset-liability problem under mean-variance framework with probability constraints. When there is no liability, Theorem 3.3 reduces to Proposition 2 in Cui-Li-Li [5]. According to (12), we can also derive the variance term as Theorem 3.4. Assume that the returns of assets and liability are correlated at every period. Then the efficient frontier of problem (3) is given by  Table 1 presents the expected values, standard deviations and correlation coefficients of the annual return rates of these three indices. We consider 5 time periods and annual risk free rate 5% (s t = 1.05). Assume that the investor has initial wealth x 0 = 3, initial liability l 0 = 1, a trade-off parameter w = 1. Furthermore, for t = 0, 1, 2, 3, 4, assume that the probability a t = 0.1, the correlation of assets and the liability is ρ = (ρ 1 , ρ 2 , ρ 3 ), where ρ i = Cov(q t , P i t ) Var(q t ) Var(P i t ) is the correlation coefficient of the i-th asset and the liability. This means 4.1. Correlation example. In this subsection, assume that the returns of the assets and liability are correlated with ρ = (ρ 1 , ρ 2 , ρ 3 ) = (−0.25, 0.5, 0.25). Hence, By interior point algorithm of "fmincon" with the initial point λ = (0, 0, 0, 0), we can obtain λ * = (0, 0, 0, 0.4902). According to Theorem 3.3, we can derive the optimal strategy of problem (3)as follows,
It is showed from the two examples that given the first and second moments of the return rates of assets and liability, the trade-off parameter and the probability, the initial wealth and liability, we can derive directly the optimal policy according Theorem 3.3 and Remark 2. By changing the parameters a t , we can see the impact of the probability on the optimal strategy and efficient frontier which is similar as Li-Li [10]. Given the same variance, the investors with the probability control of wealth less than liability obtain smaller expected terminal surplus compared to those without.
4.3. The impact of correlation coefficient. In this example, we investigate the impact of correlation coefficient of the assets and liability which has not been mentioned in most papers. The efficient frontiers in Figure 1 is computed for ω from 0.1 to 1 with a step size 0.01. As it shows, the higher the correlation is, the better the efficient frontier. It is straightforward. Consider a single period setting and assume that the investor wants to achieve some preset expected return, which is larger than the riskfree return. Due to the linearity of expectation, the investor may choose a fixed long position on risky asset no matter the value of ρ. The variance term is computed as Var(x 1 − l 1 ) = Var(x 1 ) + Var(l 1 ) − 2Cov(x 1 , l 1 ). When the correlation coefficient is higher, the larger the covariance, the smaller the variance. Therefore, the efficient frontier of higher ρ is better. Conclusion. In this paper, we employ a mean-field formulation to solve an optimal multi-period asset-liability mean-variance portfolio selection with probability constraints. Compared with using embedding scheme to deal with the nonseparable problem, we do not need to construct a system of auxiliary problems and can strictly derive its analytical optimal strategy and efficient frontier. The numerical examples show how to use the theoretical results, and the impact of the correlation of asset and liability on the efficient frontier.