The Case for Incoherence

Imagine you are stranded on a desert island. For fresh water there are three natural springs, but it is possible one or more have been poisoned. To minimize your risk, what is your optimal strategy for drinking from the springs? You might:

• select one of the three springs at random and drink exclusively from it,
• select two of the springs at random and drink exclusively from them, or
• drink from all three springs.

It doesn’t take a rocket scientist to figure out that your best strategy is to drink from just one spring. Yet, according to a certain financial theory, the optimal strategy is the diversified approach of drinking from all three springs.

I am referring to the growing literature on “coherent risk metrics” (as an aside, here is why I don’t call these “coherent risk measures.”) The whole business got started back in 1997, when Philippe Artzner, Fred Delbaen, Jean-Marc Eber and David Heath published an article “Thinking Coherently” in Risk magazine. They defined a risk metric r as “coherent” if, for any risks A and B and any constant a, it satisfies four properties:

• monotonicity: If A ≥ B, then r(A) ≥ r(B)
• sub-additivity: r(A + B) ≤ r(A) + r(B)
• positive homogeneity: for a ≥ 0, ar(A) = r(aA)
• translation invariance: r(A + a) = r(A) – a

Think of A and B as two random losses. Monotonicity requires that, if the loss A always exceeds the loss B, r(A) should exceed r(B). Sub-additivity is the “diversification” principle. It indicates that, due to diversification or hedging between the two risks, r(A + B) ≤ r(A) + r(B). Homogeneity is about scaling. To give a financial example, the risk of holding a hundred shares of a given stock should be precisely one hundred times the risk of holding one share of that stock.

The translation invariance property may seem a little bizarre. It says, in financial terms, that if you add cash to a given portfolio, the risk should decline by the amount of cash added. Most banks sweep cash from their trading portfolios each night, but no one would claim that doing so affects the riskiness of those trading portfolios. Actually, the translation invariance property reveals more about the pedigree of the coherent risk metrics business than it does about risk. If you actually go back and read the original Artzner et al. paper, you will see it did not describe properties all risk metrics should have. Rather, it focused narrowly on risk metrics that might be used for certain capital calculations. In that context, translation invariance makes perfect sense. If a counterparty deposits margin with you, your risk exposure to that counterparty is reduced.

The original Artzner et al. paper was a modest, if unsubstantial contribution to the financial risk management literature. The authors offered no justification for their four criteria other than to claim the criteria were “a priori.” That was a copout. So why have coherent risk metrics become a hot topic, with hundreds of articles published on the topic, albeit mostly in trade rags and third-tier journals?

The US military and its defense contractors learned decades ago that, to get a weapons system funded, a sexy name is essential. What meat-and-potatoes congressman could say “no” to a missile called AMRAAM. Never mind what the acronym stands for, it just shouts masculinity. What about “look down, shoot down radar” or “Apache helicopter?” The same holds true in finance. Previous authors had proposed criteria that risk metrics should satisfy, both in finance and in other disciplines—but they didn’t offer evocative names for their criteria. The Artzner et al. paper offered little that was new, except for the evocative name “coherent risk metric.” What meat-and-potatoes risk manager wants to be caught using an incoherent risk metric? Heaven forbid!

Somehow the message got lost that coherent risk metrics are applicable for only a narrow range of applications related to capital calculations. The message that people heard was that there are two types of risk metrics, those that are coherent and those that are incoherent—and value-at-risk (VaR) is incoherent. A new risk metric, called expected tail loss (ETL), was trotted out as a coherent alternative to VaR. Never mind that its convoluted definition is incomprehensible to most board members and that it is extremely sensitive to assumptions about tail behavior. Mercifully, ETL has not been widely embraced by practitioners.

Mostly, coherence enthusiasts ignored the nettlesome translation invariance property in their writings, and they treated the monotonicity and homogeneity properties as trivial. Had they not done so, they might have noticed that volatility—good old standard deviation of return—is incoherent. Instead, they focused on the sub-additively property—a property that VaR can fail to satisfy in contrived circumstances. Today, “sub-additive” and “coherent” are essentially synonyms.

At this point, you may be thinking that Artzner et al. would step forward and clear up the misunderstandings about their modest brainchild. Perhaps they could publish a popular article clarifying the limited scope of coherent risk metrics. Well, they did publish a follow-up article in 1999, but it didn’t clear up misunderstandings. It appeared in the recently-launched journal Mathematical Finance. Rather than point out to enthusiasts that their coherent risk metrics are suitable for only a narrowly-defined category of financial applications, they opened with the extravagant assertion

We provide in this paper a definition of risks (market risks as well as non-market risks) and present and justify a unified framework for the analysis, construction and implementation of measures of risk.

Anyone reading this might think their framework is not only applicable to all financial risks but perhaps even risks associated with crime prevention, aviation safety, hurricanes, etc. That is what the phrases “market risks as well as non-market risks” and “unified framework” imply. A careful reading of the paper, however, reveals the authors again sticking to a narrow category of applications related to capital. I doubt that many of the coherence enthusiasts who have cited the paper over 450 times actually read much of it. It is 22 pages of tedious blather, and enthusiasts don’t seem to have gotten the point

Buried deep in their pretentious prose, Artzner et al. are still saying, more or less:

Coherent risk metrics satisfy four criteria that the authors think are suitable for risk metrics used for certain capital calculations.

What the coherence enthusiasts continue to hear is:

All risk metrics, no matter what the application, need to be sub-additive.

This brings us full circle to my counterexample of the desert island.