Nonsensical Definitions of Economic Capital
[Technical Post] On Friday, a Vice President in our company asked me what our definition of Economic Capital was. I responded that we defined “economic capital” as the amount of capital necessary to cover unexpected losses at the 99% confidence level. That is total and complete gibberish. I have no idea what it means, I just mirror the sentence structure used by others. For examples, see Investopedia, and other sources.
Investopedia also provides a standard graphical representation, produced below:
Below I will describe why my definition is gibberish, I will contrast this to what we are really trying to say, and I’ll close by saying that this is more than a semantics problem.
The term “confidence level” is a frequentist term. It means the level at which the percentage of all samples will contain the true mean. In the frequentist world, the true mean is a fixed parameter than happens to be unknown. Therefore, when we say that we are holding risk capital equal to the 99% confidence level, all we are saying is that we are holding the amount of money that corresponds to the sampling process underlying the frequentist approach. We have said nothing about risk, and we have said nothing about the probability of ruin.
Therefore, any use of frequentist terminology should be expunged from discussion of “economic capital”. The concept of “economic capital” requires a theory of risk; a theory of risk, in turn, requires statements of uncertainty. This can only be achieved through a Bayesian approach.
The good news is that most modern approaches to economic capital don’t actually use frequentist methods that they borrow language from. Most effectively construct something akin to posterior distributions from which statements about probabilities can be made. Indeed, the internal models I’ve constructed for my own company are of this ilk; I borrow frequentist language and create gibberish statements, as above, not because this is the language I would prefer, but because this is the language of the field and the language that, even if wrong, seems to be understood by my audience. My audience, like most of humanity, has the ability to effortlessly mix frequentist and Bayesian approaches; unfortunately for me, that just makes me dizzy.
The downside of spewing gibberish, however, is that some of the risks inherent with modeling a posterior distribution get lost in the shuffle of frequentist mindsets. This means Executives lose understanding, and opportunities are lost. We rarely set clear prior belief structures with data driven updating rules, meaning that we miss the opportunity to improve and evolve our understanding of organizational risk. This means we force-fit a distribution of future conceivable outcomes and presume that it is a valid posterior distribution given our often unstated prior beliefs; this means we risk making hidden and unintentional assumptions. This could, in reality, exacerbate risk.
Perhaps most damning of all, the above approach translates risks into something scalable relative to the mean, and uses language that pretends to inform the likelihood of tail events, allowing you to bypass the thorniest of the real risk management issues. Specifically, the language above leads you to ignore discussion of the quality of the variance parameterization in the model, let alone giving you the framework to think about possible variance caused by unknowns. These items — estimation of the estimable portion of variance and the provision for the unestimable portion of variance — are, in my opinion, critical to setting appropriate economic capital targets. But the frequentist language and classical training in statistics leads us to ignore these issues.
Sometimes, I feel like Dwight Shrute screaming about right turns. I wish I could be 99% confident, like Michael Scott.
 I linked to Shim, et al, because they were rather straightforward in their paper: “More specifically, economic capital can be defined as the difference between the expected loss and a worst tolerable loss at a selected confidence level.” (empahsis added)