A multistage model: asset–liability management

The best way to introduce multistage stochastic models is a simple asset–liability management (ALM) model.²⁴ We have an initial wealth W₀, that should be properly invested in such a way to meet a liability L at the end of the planning horizon H. If possible, we would like to own a terminal wealth W_H larger than L; however, we should account properly for risk aversion, since there could be some chance to end up with a terminal wealth that is not sufficient to pay for the liability, in which case we will have to borrow some money. A nonlinear, strictly concave utility function of the difference between the terminal wealth W_H, which is a random variable, and the liability L would do the job, but this would lead to a nonlinear programming model. As an alternative, we may build a piecewise linear utility function like the one illustrated in Fig. 13.10. The utility is zero when the terminal wealth W_H matches the liability exactly. If the slope r penalizing the shortfall is larger than q, this function is concave (but not strictly).

The portfolio consists of a set of I assets. For simplicity, we assume that we may rebalance it only at a discrete set of time instants t = l, …, H − 1, with no transaction cost; the initial portfolio is chosen at time t = 0, and the liability must be paid at time H. Time period t is the period between time instants t − 1 and t. In order to represent uncertainty, we may build a tree like that in Fig. 13.11, which is a generalization of the two-stage tree of Fig. 13.9. Each node n_k in the tree corresponds to an event, where we should make some decision. We have an initial node no corresponding to time t = 0. Then, for each event node, we have two branches; each branch is labeled by a conditional probability of occurrence, P(n_k | n_i), where n_i = a(n_k) is the immediate predecessor of node n_k. Here, we have two nodes at time t = 1 and four at time t = 2, where we may rebalance our portfolio on the basis of the previous asset returns. Finally, in the eight nodes corresponding to t = 3, the leaves of the tree, we just compare the terminal wealth with the liability and evaluate the utility function. Each node of the tree is associated with the set of asset returns during the corresponding time period. A scenario consists of an event sequence, i.e., a sequence of nodes in the tree, along with the associated asset returns. We have 8 scenarios in Fig. 13.11. For instance, scenario 2 consists of the node sequence (n₀, n₁, n₃, n₈). The probability of each scenario depends on the conditional probability of each node on its path. If each branch at each node is equiprobable, i.e., the conditional probabilities are always , each scenario in the figure has probability , for s = 1, …, 8. The branching factor may be arbitrary in principle; the more branches we use, the better our ability to model uncertainty; unfortunately, the number of nodes grows exponentially with the number of stages, as well as the computational effort.

Fig. 13.11 Scenario tree for a simple asset–liability management problem.

At each node in the tree, we must make a set of decisions. In practice, we are interested in the decisions that must be implemented here and now, i.e., those corresponding to the first node of the tree; the other (recourse) decision variables are instrumental to the aim of devising a robust plan, but they are not implemented in practice, as the multistage model is solved on a rolling-horizon basis. This suggests that, in order to model the uncertainty as accurately as possible with a limited computational effort, a possible idea is to branch many paths from the initial node, and less from the subsequent nodes. Each decision at each stage may depend on the information gathered so far, but not on the future; this requirement is called a nonanticipativity condition. Essentially, this means that decisions made at time t must be the same for scenarios that cannot be distinguished at time t.²⁵ To build a model ensuring that the decision process makes sense, there are two choices:

• We can introduce a set of decision variables , representing wealth allocated to asset i at time t on scenario s; we should force decision variables to take the same value when appropriate, by writing explicit nonanticipativity constraints for scenarios that cannot be distinguished at time t.
• We can associate decision variables with nodes in the scenario trees and write the model in a way that relates each node to its predecessors.

We will illustrate the second alternative in detail, using the following numerical data:

• The initial wealth is 55.
• The target liability is 80.
• There are two assets, say, stocks and bonds; hence, I = 2.
• In the scenario tree of Fig. 13.11 we have up- and downbranches; in the (lucky) upbranches, total return is 1.25 for stocks and 1.14 for bonds; in the (bad) downbranches, total return is 1.06 for stocks and 1.12 for bonds. We see that bonds play the role of safer assets here. We also see that returns are a sequence of i.i.d. random variables, but more realistic scenarios can be defined.
• The reward rate q for excess wealth above the target liability is 1.
• The penalty rate r for the shortfall below the target liability is 4.

Let us introduce the following notation:

• N is the set of event nodes; in our case
• Each node n ∈ N, apart from the root node n₀, has a unique direct predecessor node, denoted by a(n): for instance, a(n₃) = n₁.
• There is a set S ⊂ N of leaf (terminal) nodes; in our casefor each node s ∈ S we have surplus and shortfall variables  and , related to the difference between terminal wealth and liability.
• There is a set T ⊂ N of intermediate nodes, where portfolio rebalancing may occur after the initial allocation in node n₀; in our casefor each node n ∈ {n₀} ∪ T there is a decision variable x_in, expressing the money invested in asset i at node n.

With this notation, the model may be written as follows:

where R_i,n is the total return for asset i during the period that leads to node n, and π^s is the probability of reaching the terminal node s ∈ S; this probability is the product of all the conditional probabilities on the path that leads from root node n₀ to leaf node s. This is an LP model that may be easily solved by the simplex algorithm, resulting in the solution of Table 13.2. We may notice that in the last period the portfolio is not diversified, since the whole wealth is allocated to one asset, and we should wonder if this makes sense. Actually, it is a consequence of two features of this toy model:

• We are approximating a nonlinear utility function by a piecewise linear function, and this may imply “local” risk neutrality, so that we only care about expected return; we should use either a nonlinear programming model or a more accurate representation of utility with more linear pieces.Table 13.2 Investment strategy for a simple ALM problem.
• The scenario tree has a very low branching factor, and this does not represent uncertainty accurately.

However, the portfolio allocation in the last time period is not necessarily a critical output of the model: the real stuff is the initial portfolio allocation. As we pointed out, the decision variables for future stages have the purpose of avoiding a myopic policy, but they are not meant to be implemented.