MULTI-ACCOUNT OPTIMIZATION

Portfolio managers who handle multiple accounts face an important practical issue. When individual clients' portfolios are managed, portfolio managers incorporate their clients' preferences and constraints. However, on any given trading day, the necessary trades for multiple diverse accounts are pooled and executed simultaneously. Moreover, typically trades may not be crossed, that is, it is not simply permissible to transfer an asset that should be sold on behalf of one client into the account of another client for whom the asset should be bought.²³ The trades should be executed in the market. Thus, each client's trades implicitly impact the results for the other clients: The market impact of the combined trades may be such that the benefits sought for individual accounts through trading are lost due to increased overall transaction costs. A robust multi-account management process should ensure accurate accounting and fair distribution of transaction costs among the individual accounts.

One possibility to handle the effect of trading in multiple accounts is to use an iterative process, in which at each iteration the market impact of the trades in previous iterations is taken into account.²⁴ More precisely, single clients' accounts are optimized as usual, and once the optimal allocations are obtained, the portfolio manager aggregates the trades and computes the actual marginal transaction costs based on the aggregate level of trading. The portfolio manager then reoptimizes individual accounts using these marginal transaction costs, and aggregates the resulting trades again to compute new marginal transaction costs, and so on. The advantage of this approach is that little needs to be changed in the way individual accounts are typically handled, so the existing single-account optimization and management infrastructure can be reused. The disadvantage is that most generally, this iterative approach does not guarantee a convergence (or its convergence may be slow) to a “fair equilibrium,” in which clients' portfolios receive an unbiased treatment with respect to the size and the constraint structure of their accounts.²⁵ The latter equilibrium is the one that would be attained if all clients traded independently and competitively in the market for liquidity, and is thus the correct and fair solution to the aggregate trading problem.

An alternative and more comprehensive approach is to optimize trades across all accounts simultaneously. O'Cinneide, Scherer, and Xu²⁶ describe such a model and show that it attains the fair equilibrium we mentioned previously.²⁷ Assume that client k's utility function is given by u_k, and is in the form of a dollar return penalized for risk. Assume also that a transaction cost model τ gives the cost of trading in dollars, and that τ is a convex increasing function.²⁸ Its exact form will depend on the details of how trading is implemented. Let t be the vector of trades. It will typically have the form , that is, it will specify the aggregate buys and the aggregate sells for each asset i = 1, …, N, but it may also incorporate information about how the trade could be carried out.²⁹

The multi-account optimization problem can be formulated as

where w_k is the N-dimensional vector of asset holdings (or weights) of client k, and C_k is the collection of constraints on the portfolio structure of client k. The objective can be interpreted as maximization of net expected utility, that is, as maximization of the expected dollar return penalized for risk and net of transaction costs.

The problem can be simplified by making some reasonable assumptions. For example, it can be assumed that the transaction cost function τ is additive across different assets, that is, trades in one asset do not influence trading costs in another. In such a case, the trading cost function can be split into more manageable terms, that is,

where is the cost of trading asset i as a function of the aggregate buys and sells of that asset. Splitting the terms further into separate costs of buying and selling, however, is not a reasonable assumption, because simultaneous buying and selling of an asset tends to have an offsetting effect on its price.

To formulate the problem completely, let be the vector of original holdings (or weights) of client k's portfolio, W_k be the vector of decision variables for the optimal holdings (or weights) of client k's portfolio, and η_k,i be constants that convert the holdings (or weight) of each asset i in client i's portfolio w_k,i to dollars, that is, η_k,iw_k,i is client k's dollar holdings of asset i.³⁰ We also introduce new variables to represent the an upper bound on the weight of each asset client k will buy:

The aggregate amount of asset i bought for all clients can then be computed as

The aggregate amount of asset i sold for all clients can be easily expressed by noticing that the difference between the amounts bought and sold of each asset is exactly equal to the total amount of trades needed to get from the original position to the final position w_k,i of that asset:³¹

Here and are nonnegative variables.

The multi-account optimization problem then takes the form

O'Cinneide, Scherer, and Xu studied the behavior of the model in simulated experiments with a simple model for the transaction cost function, namely one in which

where t is the trade size, and θ and γ are constants satisfying θ ≥ 0 and γ ≥ 1.³² θ and γ are specified in advance and calibrated to fit observed trading costs in the market. The transaction costs for each client k can therefore be expressed as

O'Cinneide, Scherer and Xu observed that key portfolio performance measures, such as the information ratio (IR),³³ turnover, and total transaction costs, change under this model relative to the traditional approach. Not surprisingly, the turnover and the net information ratios of the portfolios obtained with multi-account optimization are lower than those obtained with single-account optimization under the assumption that accounts are traded separately, while transaction costs are higher. These results are in fact more realistic, and are a better representation of the post-optimization performance of multiple client accounts in practice.