Game theory plays well for capital management

The basic objective of every bank is to find an optimal business strategy that maximises return on capital (ROC). To this end, banks will allocate more capital to desks that generate the highest ROC, while those with lower ROC receive a smaller share of available capital. This is an intuitive and seemingly sensible solution.

But the authors of this month’s paper, Reduced-form capital optimisation, argue that ranking business units by ROC might not result in an optimal allocation of capital. The approach ignores the correlation between businesses and will deliver the optimal allocation only if the correlation is zero – an assumption for which there is little evidence in the real world.

The problem is further complicated by the Basel III capital rules, which are primarily based on two ratios: risk-weighted assets (RWAs) and leverage balance sheet (LBS). The minimum capital requirement for each legal entity under the same parent bank is the greater of the two.

To optimise its ROC, a legal entity must hold RWA and LBS capital in equal measure. The selection of RWA or LBS, as directed by the greater-of-the-two rule, introduces non-linearities that are difficult to deal with in an optimisation context.    

“This is a huge, unsolved problem for banks,” says Yadong Li, managing director of quantitative analytics at Barclays, and one of the authors of the paper. “The bank as a whole ultimately needs to deploy the resources to different business units. This is a key part of senior management’s job.”

Dimitri Offengenden, a Tel Aviv-based managing director of quantitative strategy at Barclays, joined the bank in 2017 specifically to work on the capital allocation problem. He consulted with Li, who has worked on related issues in the past.

This is a huge, unsolved problem for banks

Yadong Li, Barclays

“I suggested we look at Shapley allocation, which is a natural way to solve the greater-of-two problem” says Li.

Shapley allocation is well known in game theory and has been widely applied in economics and business decision making. It attributes a value to the contribution of each agent in a system where agents co-operate and share the costs and gains of their activity. In essence, it measures the marginal contribution of each agent by observing the difference the agent’s presence or absence makes to the activity. Based on that, a portion of the shareable pie is allocated to each agent. In the case of a bank, the agents are individual business units and the pie is the overall capital stock.

Offengenden, Li and Jan Burgy, a quant strategist at Barclays, realised that regressing the overall allocation of capital on the selection of RWAs and LBS obtained from the application of the Shapley method resulted in an almost perfect approximation. The resulting R-squared – a statistical measure that indicates how well one variable explains another – was an enormous 98–99%.

Reduced-form capital optimisation

The key innovation of the paper is the discovery of an accurate linear approximation of the Shapley allocation of the maximising function of RWA and LBS capital to business units.

The finding laid the groundwork for a further simplification. “Once we verified the linear relationship was real and robust, we wanted to be more ambitious. With this linear relationship we could formulate the entire capital optimisation in a very simple form and turn it into a classic mean-variance optimisation” says Li.

The mean-variance representation leads to a reduced-form solution to the capital optimisation problem. In their framework, the mean is the ROC and the variance is a measure of the cost of changing the allocation of capital. The approach is made more intuitive when viewed in the context of an efficient frontier. The business units that capital should be allocated to are identified just like securities in a mean-variance approach to portfolio selection.

“It is clearly an interesting alternative to standard allocation methods,” says the head of internal risk capital methodology at a major European bank. He says the paper is even more relevant because it investigates the matter “from the group perspective of a very large institution, where there are very different and heterogeneous contributions from RWAs”. 

This research builds on an earlier paper, Organising the allocation, published in Risk.net in 2016, of which Li is one of the authors. That paper addressed another issue associated with the rule-of-thumb approach commonly used to allocate capital. The calculation of ROC at the desk level isn’t straightforward. For regulatory and reporting purposes, capital is commonly computed at the bank level, and there is no universally accepted way to allocate it to individual desks. The earlier paper introduced a more computationally efficient variation of the Shapley method, called the constrained Aumann-Shapley method – which is also used in the most recent paper – that allows for an efficient allocation of capital between business units.