Optimal capital allocation principles

Winner of the 2011 Lloyd's Science of Risk Prize, in the category of Insurance Markets and Operations.

This paper develops a unifying framework for allocating the aggregate capital of a financial firm to its business units.

Introduction to new research by Andreas Tsanakas, Senior Lecturer in Actuarial Science at Cass Business School.

Financial firms, such as insurance companies or banks, need to hold a level of safely invested risk capital to protect themselves against unexpected losses.  It is common practice to allocate the total required capital for the portfolio to its constituent parts, e.g. lines of insurance business. Capital allocation, often linked to return-on-equity arguments, provides a useful method for assessing and comparing the performance of different sub-portfolios. Allocating capital may also help to identify areas of risk consumption and support decision-making concerning business expansions, reductions or even eliminations.

Because of portfolio diversification effects, there is no single way in which to carry out such a capital allocation exercise. Some of the methods used in practice or proposed in the literature are underpinned by very different arguments, while others are remain quite arbitrary. In an insurance world dominated by Solvency II, linking capital allocation to internal processes, such as performance measurement, pricing, and portfolio optimisation, is an emerging tough requirement. It is hard to envisage such exercises being successful, when the methods used are not explicitly linked to management’s thinking and formulated risk appetite.

Our own contribution to that debate is to, first, provide a  unifying framework for capital allocation methods, which covers most known approaches; and, second, give a business-driven interpretation of these methods, thus enabling the formulation of an explicit link between risk appetite and decision-making.

Our approach follows two fundamental principles. First, we claim that the aggregate capital that needs to be held for the portfolio should be derived from specific adverse scenarios that may materialise in the future. Different firms, and for different applications, may require focus on different ranges of such scenarios; for example these may be scenarios under which financial markets perform poorly, or scenarios where the firm’s own idiosyncratic loss is high. Scenarios may be formulated, at either the level of the portfolio or at that of a business line, reflecting the diverging ways in which diversification may be considered. Depending on what the risk management priorities are in the context of a given exercise, scenarios are assigned individual weights, and the total capital is calculated as the weighted average of losses arising from these scenarios.

Once the adverse scenarios of interest and their weights have been fixed, we apply the second principle in our approach, that is, for each line of business, the allocated capital should be as close as possible to the potential loss. Of course “closeness” has to be formally defined, and we do this by using mathematical notions of distance, in particular the quadratic and absolute deviations. What is crucial is that the distance between capital on the one side and risk on the other, is calculated again by weighting adverse scenarios , in the same way as is done in the calculation of aggregate capital. Thereby, capital allocation is derived consistently with the management preferences that underlie the aggregate capital levels.

The optimisation problems from which capital allocations derive are solved in their general form. Explicit allocation formulas are obtained, which, by their scenario-based structure, can be easily implemented via Monte-Carlo simulation. When focusing on specific cases, we find that nearly every capital allocation method encountered in the literature can be seen as a special case of our approach. That includes methods as diverse as marginal/Euler-type allocations; allocations based on the value of the default option; allocations driven by market-consistently calculated solvency ratios; and VaR-based allocations, where diversification is recognised by requiring a lower confidence level for business lines than for the portfolio.

In the case that the aggregate capital is exogenously given and not set as part of the allocation exercise, capital can still be optimally allocated according to the above arguments and similar allocations emerge; it turns out the difference between the theoretical and the actually available level of capital is allocated on a weighted proportional basis.

The generality of our approach allows easy derivation of new capital allocation methods, given appropriate formulations of scenario weights. Nonetheless, we believe that it is less important to add more methods to an already crowded landscape, than to explore what given methods mean and on which implicit assumptions they rest. If each allocation method is an answer to a specific business question, our approach allows risk managers to clearly formulate that question and then allocate capital in a consistent way.