Allocating Capital: Another Tactic
A Review of Michael Kalkbrener's "An Axiomatic Approach to Capital Allocation"

Note from Stephen Philbrick: Capital allocation, or the allocation of the cost of capital, is a subject of great interest to me, as past columns will attest. I thank Glenn for allowing me to use his review as this issue's column. The entire concept is important, but Glenn illustrates one particular item I'll emphasize - when a particular allocation rule violates axioms, one shouldn't automatically discard the rule. Use it as a framework to seriously examine and understand the implications of the axioms. In some cases, it will suggest a refinement to the axioms, rather than a change to the rule.

Supplement 1 (.pdf)
Supplement 2 (.xls)

An Axiomatic Approach to Captial Allocation
by Michael Kalkbrener

Reviewed by
Glenn Meyers

Allocating capital has always been a controversial topic among actuaries. See, for example, Gary Venter's article in the February, 2002 Actuarial Review titled "Allocating Surplus - Not." After years of debate I think that a common (if not prevailing) view of capital allocation is that is a useful tool to manage an insurer's portfolio of risks. The general idea behind capital allocation is that those lines that contribute the most risk to an insurer's portfolio get allocated the most capital. One of the reasons for the controversy is that most actuaries do not feel comfortable with the mathematical properties of some of the proposed formulas that take risk into account. A common problem is that the sum of the allocated capitals does not equal the total capital. A second related problem is that the allocated capital may depend on the order in which it enters the insurer's portfolio.

Enter Michael Kalkbrener. His paper, titled "An Axiomatic Approach to Capital Allocation," which appears in July 2005 issue of Mathematical Finance, proposes a set of axioms that I think most actuaries would consider reasonable and then shows that these axioms determine a unique capital allocation formula! Let's look at the axioms.

Let Χ and Υ denote two insurance portfolios. Let Λ (Χ, Υ) denote the part of the capital of Υ that is allocated to Χ. Let ρ be a positively homogeneous and subadditive measure of risk. (See my August 2002 Actuarial Review article on "Coherent Measures of Risk" for definitions.) We also require that Λ (Χ, Χ) = ρ (Χ).

The three axioms are called (1) linear, (2) diversifying and (3) continuous. The linea axiom guarantees that the allocated capital "adds up." The "diversifying" axiom guarantees that a segment of business' allocated capital by is no larger than its standalone capital. The "continuous" axiom guarantees that small changes in a segment's portfolio will not make large changes in is allocated capital.

Kalkbrener shows that for every positively homogeneous and subadditive risk measure, ρ , there is a unique linear, diversifying and continuous capital allocation Λ. He also shows that Λ can be calculated by the directional derivative:

Here is how this formula works on a simple example. Let Υ = Χ₁+Χ₂ be random variables generated on a finite set of scenarios generated by a simulation. Let the capital be determined by ρ(Υ) = max(Υ). To allocate capital to each X_i) we first find the scenario (χ₁,χ₂) where Υ takes on its maximum. (To keep the math simple in this example, I am considering only the case where (χ₁,χ₂) is unique.) Thus, (Υ) = χ₁+χ2. Then for sufficiently small ε, max(Χ₁+Χ₂+ εΧ_i) = χ₁+χ₂+ εχ_i and thus:

Kalkbrener gives a general version of this example using coherent measures of risk, of which the max(.) measure is a special case.

In this example, the capital allocated to x_i is equal to its contribution to the maximum loss scenario. Those who have been following this issue will recognize this as the same principle that David Ruhm and Don Mango in a number of their papers that describe what is coming to be called the RMK algorithm.

If we use the risk measure ρ(Χ) = Ε(Χ)+Τ•Std(Χ), Kalkbrener shows that his allocation formula gives the covariance allocation formula described in separate papers by Dan Gogol, Don Mango and Gary Venter.

Let's now consider an alternative approach to capital allocation, which I will call economic allocation. Suppose you are an insurer with a given portfolio of business. Suppose further that you determine your capital using a subadditive measure of risk and you measure your profitability in terms of the return on your investment. Economics 101 advises you to improve your profitability by increasing your exposure in lines that give you the highest marginal return on your investment and reduce your exposure in the lines that give you the lowest marginal return. Over time this strategy results in an insurer having equal marginal returns on its investment for all lines of insurance. If you choose to express this strategy in the language of capital allocation, you will allocate capital in proportion to the marginal capital for each line of insurance. See my paper "The Economics of Capital Allocation" for a more mathematically rigorous version of this argument. Unfortunately it turns out that economic allocation is not equivalent to axiomatic allocation. I have been able to construct an example where the two approaches yield different answers.

Which axioms are violated by economic allocation? The answer is found by applying the steps of Kalkbrener's derivation to an economic allocation formula. It turns out that the diversifying axiom is violated. It is possible for a standalone portfolio of risks to increase its allocated capital when considered as a part of a larger portfolio of risks.

My personal preference is for the economic formula. I think capital allocation should be driven by economic objectives, and not mathematical properties. But having participated in these debates over the years I recognize the power that the axiomatic approach holds over many of my actuarial colleagues. Based on the examples I have constructed to date, I suspect that the differences between the two formulas would not excite my more "practical" minded colleagues. In spite of my personal preferences, I consider this paper to be one of the finest I have read on this subject and I hope that many CAS members will read it. Final note - The examples referred to in this article are available in a supplement that is on the CAS website along with the web version of this article. There is also an accompanying spreadsheet.

Final note: The Web version of this AR Brainstorm's column is expanded to include an exact statement of the axioms and more details on the examples that I refer to. There is also an accompanying spreadsheet.