Allocating capital should be beautifully simple. It just needs a slight adjustment to reality

Most financial institutions are somehow constrained by a budget of risk or capital requirements. Since this in an overall requirement for the business as a whole, it can be hard to reflect the constraint in everyday decisions. Read on to learn how maths can help a lot, and how reality can give you a little extra work.

So, why care about capital allocation?

Capital allocation (in the sense used here) is all about creating organisational ownership of the risks taken or capital requirements incurred. Income and cost are traditionally allocated to business activities in order to push the incentive to optimise out to the individual decision makers. The same can, and should, be done with capital consumption or risk. It is not uncommon for a financial institution to have capital costs of the same magnitude as operational cost, so why should it be treated any differently?

The hard part of allocating capital or risk is, that the measures are often complicated and non-linear. They can rely on simulations and involve diversification between risk types, and this makes them hard for the individual business users to understand. A good allocation can help this understanding if it has these properties:

  • Completeness: A capital allocation model is no use if the allocated capital does not sum up to the total capital it tries to allocate.
  • Business users should be able to understand what drives their numbers without understanding the whole allocation model. This does not mean that the model cannot be complex, you just need to hide it from the users.
  • The same risk taken in different parts of the organisation should have the same allocated capital. If not, it will erode buy-in for the allocation model.

The maths – beautiful as always

Luckily, you can all get this from a mathematical theorem. No matter how complicated a risk or capital measure you have, it basically only requires that when all exposures are doubled, the total risk doubles. This seems reasonable – if it does not hold, you probably have a more fundamental problem with your risk measure, that you have to solve first.

When it holds, there is a way of constructing weights for each type of exposure, so that

Risk = weight1 * Exposure1 + weight2 * Exposure2 + … + weightN * ExposureN

The weights can be kept constant for the whole business, and to get the risk allocated to any subdivision of the business, you use the exposures of that subdivision. The weights depend on all the exposures at once, so in a sense this is just a fancy rewriting of the original risk measure. But it gives you the properties you want

  • Since the exposures over business subdivisions sum up to the exposures of the whole business, the allocation is complete
  • Business users only have to understand that there are weights that they must use. The complexity is hidden away in the weights.
  • An exposure will contribute the same allocated capital because the weight does not depend on the business unit where it resides. There is a business-wide ‘price of risk’.

The reality – messy as always

Up to now, I did not talk about what the risk or capital measure actually was. In reality, this can be all sorts of things: Formula or simulation; Value-at-Risk, Expected Shortfall, or standard deviation; internal or regulatory. Unfortunately, every application of math to the real world comes with a bit of hazzle.

The typical hazzle involves some element of the original risk measure, or the way it is used, that makes the allocation complex over time or gives wrong business incentives. Solving the problem is a matter of giving up completeness, simplicity, or fairness of the allocation. Which one to sacrifice depends on the concrete business. In my experience, people will most often give up completeness, as long as the mismatch is not too big.

Here are three real-world examples.

Example 1: Value-at-Risk simulations

Value-at-Risk is often computed by running a range of randomised scenarios and computing the losses in each scenario to produce a loss distribution. If you run 10,000 scenarios and want a 1% Value-at-Risk, you just find the 100th-worst loss among your simulated losses.

When doing this, the weights in the risk allocation are the unit losses for each exposure in the 100th-worst scenario. This happens because tiny changes in exposures will not change the fact that this scenario is number 100.

For many purposes, this will be alright, but if you have exposures that are non-linear, or often change directions, there are often many different ‘ways’ of producing a 1% loss. In each of these ways, weights will differ significantly, but it is arbitrary which way of losing money caused exactly the 100th-worst scenario. It will also change from time to time, giving the decision makers in your business wildly varying weights and allocated risk.

A popular solution is to use some kind of smoothing over the scenarios close to the 100th-worst, instead of picking exactly that one. On the con side, you will ruin completeness and will have to continually supervise that you are not smoothing too much.

Example 2: Solvency II Standard Approach

I once worked for a major Danish life insurer on a project to allocate the Solvency Capital Requirement under the Solvency II Standard Approach. At first sight, this looks like a very benign problem if you know the Standard Approach: it is a (big) closed formula expression, taking as input various exposures to market and underwriting risks.

The issue was that some risks can go both ways. You can have both positive and negative exposure to interest rates. The solution in the Standard Approach is to use the larger of the ‘up’ and ‘down’ risks. This is completely fine if the business is consistently exposed to, say, interest rate decreases. In that case, the weight for the exposure to a government bond would be negative, because buying the bond would partially hedge the risk.

This company, however, was running a pretty tight interest hedging programme, so the direction of the interest rate exposure would change often. And every time the overall interest rate exposure changed sign, all the weights on fixed income instruments would change signs too. Very impractical to the people whose performance was to be measured this way.

There could be several solutions to this. You could choose to ignore interest rate risk entirely, overwrite the weights with zeros, or something else. Again, all these solutions give up completeness of the allocation, and would require that you control interest rate risk in another way.

Example 3: Window dressing

Most allocation of risk relates to regulatory requirements or external reporting. The frequency of external or regulatory reporting is often quarterly or annual, while the business is conducted on a shorter timescale. Consider, as an example, a measure of market risk that is reported to authorities quarterly, but which can change daily due to portfolio changes.

In principle, this gives the business an incentive to window dress: running the business at one risk level between reporting dates, and at another risk level close to reporting dates. In reality, there are often legal and reputational reasons why a business would not want to do this.

Problem is, if you only charge business units for their risk on reporting dates, you forward the incentive to window dress to every decision-maker in the business. Do they care as much about the legal and reputational repercussions of window dressing as the whole business does?

The most straightforward solution is to charge the business units for their use of risk on a higher frequency than the external reporting. Ideally, it should be on the timescale on which the business units can meaningfully change their exposures. To do that, you must be able to measure at a higher frequency, which would be good governance anyway. And you give up completeness of the allocation.

A few last words

Implementing a capital allocation tends to be a process that is different for every business, and without standardised methodologies to pick. When you jump into it, be sure to know what corners you want to cut, and which you want to make as sharp as possible. Good luck!

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