Asset–liability management with transaction costs

To give the reader an idea of how to build nontrivial financial planning models, we generalize a bit the model formulation of the previous section, in order to account for proportional transaction costs. The assumptions and the limitations behind this extended model are the following:

• We are given a set of initial holdings for each asset; this is a more realistic assumption, since we should use the model to rebalance the portfolio periodically, according to a rolling-horizon strategy.
• We take proportional (linear) transaction costs into account; the transaction cost is a percentage c of the traded value, for both buying and selling an asset.
• We want to maximize the expected utility of the terminal wealth.
• There is a stream of uncertain liabilities that we have to meet.
• We do not consider the possibility of borrowing money; we assume that all of the available wealth at each rebalancing period is invested in the available assets; actually, the possibility of investing in a risk-free asset is implicit in the model.
• We do not consider the possibility of investing new cash at each rebalancing date (as would be the case, e.g., for a pension fund).

Some of the limitations of the model may easily be relaxed. The important point we make is that when transaction costs are involved, we have to introduce new decision variables to express the amount of assets (number of shares, not the monetary value) held, sold, and bought at each rebalancing date. We use a notation which is similar to that used in the previous ALM formulation:

• N is the set of nodes in the tree; n₀ is the root node.
• The (unique) predecessor of node n ∈ N\{n₀} is denoted by a(n); the set of terminal nodes is denoted by S; as in the previous formulation, each of these nodes corresponds to a scenario, which is the sequence of event nodes along the unique path leading from n₀ to s ∈ S, with probability π^s.
• T = N\({n₀} ∪ S) is the set of intermediate trading nodes.
• Lⁿ is the liability we have to meet in node n ∈ N; liabilities are node dependent and stochastic.
• c is the percentage transaction cost.
•  is the initial holding for asset i = 1, …, I at the root node.
•  is the price for asset i at node n.
•  is the amount of asset i purchased at node n.
•  is the amount of asset i sold at node n.
•  is the amount of asset i we hold at node n, after rebalancing.
• W^s is the wealth at terminal node s ∈ S.
• u(w) is the utility for wealth w; this function is used to express utility of terminal wealth.

On the basis this notation, we may write the following model:

The objective (13.28) is the expected utility of the terminal wealth; if we approximate this nonlinear concave function by a piecewise linear concave function, we get an LP problem (as we did in Section 12.4.7). Equation (13.29) expresses the initial asset balance, taking the current holdings into account; the asset balance at intermediate trading dates is taken into account by Eq. (13.30). Equation (13.31) ensures that enough cash is generated by selling assets in order to meet the liabilities; we may also reinvest the proceeds of what we sell in new asset holdings; note how the transaction costs are expressed for selling and purchasing. Equation (13.32) is used to evaluate terminal wealth at leaf nodes; note here that we have not taken into account the need to sell assets in order to generate the cash required by the last liability; but this would make only sense if the whole fund is liquidated at the end of the planning horizon. If so, we could rewrite Eq. (13.32) as

In practice, we would repeatedly solve the model on a rolling-horizon basis, so the exact expression of the objective function is a bit debatable. The role of terminal utility is just to ensure that we are left in a good position at the end of the planning horizon.

This model can be generalized in a number of ways, which are left as an exercise to the reader. The most important point is that we have assumed that the liabilities must be met. This may be a very hard constraint; if extreme scenarios are included in the formulation, as they should be, it may well be the case that the model above is infeasible. Therefore, the formulation should be relaxed in a sensible way; we could consider the possibility of borrowing cash; we could also introduce suitable penalties for not meeting the liabilities. In principle, we could also require that the probability of not meeting the liabilities is small enough; this leads to chance-constrained formulations, for which we refer the reader to the literature.