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The Actuary The magazine of the Institute & Faculty of Actuaries

Capital ideas

Multiple approaches to capital allocation bring multifarious insights,
says Paul Johnson


Businessman using binoculars-shutterstock
Photo: shutterstock

For years, the primary use of capital models has been to produce regulatory capital figures. There are many secondary uses, however, such as economic capital, reinsurance optimisation, investment optimisation, strategic planning and capital allocation. Some firms might argue this primary/secondary split is not so clear-cut, and their capital models are so well embedded within the risk management system that several uses of the model could be considered as primary. This is particularly true if capital allocation to business lines or risks drives performance measurement, product pricing, and so on. It is interesting to consider the evolution of capital allocation that has occurred alongside the development of capital models.

Before today’s complex and complete stochastic models, typically, regulatory capital figures were derived by combining 1-in-200 (or 99.5th percentile) stresses for different risk categories (typically, underwriting, market, credit, reserving, operational and group risks) using a correlation matrix. The ensuing capital allocation often followed a simplistic pro-rata approach, whereby diversified capital was allocated to risks, products or entities within a group, based on their undiversified 1-in-200 stress.

Given the computing power and models available at the time, this approach to calculating regulatory capital (and its allocation) was entirely reasonable, but the problem was this approach did not give much insight into the drivers of capital, or the scenarios that caused the business to fail. We can draw a parallel here with producing scenarios for reverse stress tests. Furthermore, the allocation depended, as it does today, on the time horizon that was considered, with a run-off approach being common within Lloyds, compared with the one-year horizon required under Solvency II. In simple terms, this year’s problems might be different from the problems that could arise in five to 10 years’ time.

Irrespective of the details of time horizon or the breakdown of risk that we choose, the issue of looking at the problems of the business as a whole is at the crux of the matter. Management teams in some cases are used to the older approach, and feel comfortable with a pro-rata approach to capital allocation. This is consistent with a view that the primary use of the model is to produce regulatory capital. Managers intuitively understand the results of individual risk categories and the concept of an overall capital figure, but it is harder to have an intuitive feel for the allocation of capital to different business lines or risk drivers, and, related, the level of correlations between some risk pairings. 

Table 1: Illustrative allocation of required capital to risks based on diff erent risk measures
Table 1: Illustrative allocation of required capital to risks based on diff erent risk measures

Today’s challenges

There are many useful analytic capital allocation methods that exist in today’s world of fully stochastic models. Typically, we need the loss for each business line or risk and the aggregate loss as a starting point for each simulation of the model. There are a number of decisions that need to be made:

● At what level of granularity do we want to allocate capital, in order to be able to support effective management of the business?

● What is the ‘fairest’ method of allocation, given that the performance of different products, or indeed the assessment of risk appetite, may depend on this?

● Included in the above, do we want to allocate capital based on a value at risk (VaR) measure of capital, or some other measure?

● What percentile of the distribution should we focus on – that is, how far into the tail do we want to look when determining an allocation of capital to business lines or risks?

It is important to highlight the conceptual difference between the risk measure underlying the regulatory capital result, and that being used for the purpose of allocation of this overall capital. The regulatory capital under Solvency II is the 99.5th percentile VaR measured over a one-year time horizon. 

For capital allocation purposes, an alternative to the pro-rata method would consider a ‘smoothed biting scenario’, in which we take some average of the scenarios around the 99.5th percentile biting scenario to provide a single scenario from which a reasonably stable allocation of capital to business lines can be determined. Another method would consider the tail value at risk (TVaR), defined as the expected value of the loss, given that it exceeds a defined point in the loss distribution, as the starting point for allocation. If we use, say, the 98th percentile TVaR (the expected loss given that the loss exceeds the 98th percentile loss) as the base, we may end up with a very different allocation. We could also consider a slice of simulations between, for example, the 75th and 95th percentiles. This is the spread value at risk (SVaR) approach.

From this description, we can infer there are many different possible results for capital allocation, and each approach tells us something. Different allocation methods tell us “what is going wrong” in different parts of the distribution. If we were to take only one approach, we would get a skewed view and almost certainly make sub-optimal decisions. 

Another key question that arises relates to embedding of the model in decision-making. Should we focus on the 1-in-200 measure for capital allocation, and therefore decision-making, or some other level? The risk appetite of the firm may dictate that a more extreme tail is appropriate as a driver for decision-making. But, on the other hand, it is also helpful to know what is most likely to cause us to miss our business plan by, say, 50% to 100% of the planned profit. We could put strategic measures in place to mitigate all areas of the loss distribution in which we are interested, but the measures might be different and might even be in conflict with one another.

For example, purchasing expensive high-level reinsurance cover could reduce the capital allocated to peak risks, but it might jeopardise the profitability of the business plan at the mean and push the whole distribution further into loss. In fact, the profitability and return on capital on the business plan could well be improved by buying less, rather than more, reinsurance cover. The more quality management information that we have, the better our strategic decisions have a chance of being.
Table 1 above is an example of the output that might be produced on a regular basis. 

Material differences in allocation between risk measures should be fully explored. The risk categories may be subdivided further based on the level at which risks are modelled and managed. There are many different possible allocations that can be illustrated in addition to the five in the above table. The most appropriate sub-set for the business should be decided by management, taking into account their risk appetite, and this selection could be reviewed annually.

Where does this leave us?

While there is a prescribed risk measure for calculating regulatory capital, there are lots of acceptable ways to allocate this capital and we can learn different things from different approaches. The simplistic pro-rata approach over-allocates capital to minor risks that actually ‘diversify away’. Unless a minor risk is very strongly correlated to a major risk, it does not seem ‘fair’ to allocate a significant amount of capital to it. This overstatement of risk could lead to taking less risk in one area than is optimal for the return on capital. A more sophisticated method may be better, but does not necessarily provide enough insight or lead to the right decisions when considered in isolation.

Having said all that, it is important to recognise that boards will want to select a single capital allocation that can be used for performance measurement. This selection should take into account a range of allocation options, possibly blended, and should also consider the risk appetite and strategic aims of the business. Developing an improved understanding of the drivers of capital across different sections of the profit and loss distribution should help management better understand the drivers of risk to the business as a whole. This should lead to more informed decision-making around capital and strategy.

PAUL JOHNSON is director of Everlong Actuarial Services and has vast experience of the London market