PDA

View Full Version : Coherent Risk Measures


tdconrad
01-25-2008, 07:06 PM
Is anyone familiar with this term who can direct me to a high level overview of the idea? I am curious about how important the property of coherence is in a risk measure when it comes to determining capital requirements. What are the pros and cons of a risk measure that is or is not coherent?

Pete Lindquist
01-26-2008, 11:32 AM
Google found this:

http://www.wilmott.com/messageview.cfm?catid=19&threadid=7678

Pete

jraven
01-26-2008, 12:38 PM
One of the Exam 4/C study notes (available online) is an introduction to risk measures, and it defines coherence. I don't recall how much it says about why subadditivity is desirable though. The basic issue is just that diversification shouldn't make things worse, so a risk measure should preferably have risk(A+B) <= risk(A) + risk(B).

Of course VaR isn't coherent, and that still gets plenty of use due to Basel requirements.

campbell
01-26-2008, 01:36 PM
VaR isn't coherent, but it can take some really strong negative correlations for it to screw up in a weird way when you break up risks into component parts. It's a popular measure because it's much easier to explain to a board of directors than CTE or Wang's transform, and because it's simple to compute. CTE has some plusses beyond coherence, especially when you not only care that something has failed, but how severe the failure is.

The simplest explanation of coherent risk measures in SOA materials can be found here:
http://www.soa.org/files/pdf/C-25-07.pdf

As mentioned above, this paper is on the 4/C syllabus.

Glenn Meyers
01-28-2008, 03:09 PM
I put two articles in the Actuarial Review a few years agos. Here are the links.

http://www.casact.org/pubs/actrev/aug02/latest.htm

http://www.casact.org/newsletter/index.cfm?fa=viewart&id=4901

Gareth
01-28-2008, 06:34 PM
Subadditivity doesn't always hold in real life though. Consider a reinsurer that has two business units:

A) that writes low risk quota share and

B) that specializes in high risk cat cover

Is risk(A+B) <= risk(A) + risk(B) true?

I'd say no. If you isolate A and B in separate legal entities then when looking at the nasty scenarios in the tails we can easily find that B goes bust but A will be okay.

Now consider the aggregate entity A+B. A bad cat event can now drain all the capital of A and B and the shareholder is left with nothing...

jraven
01-28-2008, 09:30 PM
Subadditivity doesn't always hold in real life though. Consider a reinsurer that has two business units:

A) that writes low risk quota share and

B) that specializes in high risk cat cover

Is risk(A+B) <= risk(A) + risk(B) true?

I'd say no. If you isolate A and B in separate legal entities...

A and B are loss random variables, and risk is a functional assigning a value to those random variables. If you want to introduce a shareholder's view of limited liability into the mix -- say a maximum loss of a on A and b on B versus a joint cap of a+b -- then you have changed the random variables under consideration; it instead becomes a question about risk[(A+B) ^ (a+b)] versus risk[A ^ a] + risk[B ^ b], to which subadditivity doesn't apply.

Glenn Meyers
01-28-2008, 10:03 PM
Subadditivity doesn't always hold in real life though. Consider a reinsurer that has two business units:

A) that writes low risk quota share and

B) that specializes in high risk cat cover

Is risk(A+B) <= risk(A) + risk(B) true?

I'd say no. If you isolate A and B in separate legal entities then when looking at the nasty scenarios in the tails we can easily find that B goes bust but A will be okay.

Now consider the aggregate entity A+B. A bad cat event can now drain all the capital of A and B and the shareholder is left with nothing...

If you use a coherent measure of risk, the answer to your question is yes. Try it with the tail value at risk. It will also work with the standard deviation measure which is subadditive, but fails one of the other coherent measure axioms.

Gareth - if you still say no, I suggest that you try to come up with another measure of risk that gives you an answer that you can live with. Then see if your measure has any other problems.

tdconrad
01-29-2008, 05:44 PM
Thanks everyone for the references-- time to start reading ...

tdconrad
01-31-2008, 11:37 AM
The basic issue is just that diversification shouldn't make things worse, so a risk measure should preferably have risk(A+B) <= risk(A) + risk(B).

Could someone elaborate on this point? I agree that diversification shouldn't make thing worse, but there are certainly situations where it does (depending on how you define "worse"). Many of the papers citing the important of sub-additivity give examples of situations where two separate entities with low probabilities of ruin combine to have a higher probability of ruin-- because, when using VaR, this leads to higher capital requirements for the combined entity, sub-additivity is violated, and the VaR metric is deemed un-desirable.

But if the probability of ruin is what is most important to me, or perhaps all that is important to me, wouldn't I want to use a metric that highlights this flaw of the merger of these to entities (i.e. VaR)? It seems that by limiting ourselves to coherent risk measures, we would dismiss any indicators that show increasing risks of ruin in merger situations. Is this necessarily desirable?

Ganymede
02-01-2008, 11:16 AM
But if the probability of ruin is what is most important to me, or perhaps all that is important to me, wouldn't I want to use a metric that highlights this flaw of the merger of these to entities (i.e. VaR)? It seems that by limiting ourselves to coherent risk measures, we would dismiss any indicators that show increasing risks of ruin in merger situations. Is this necessarily desirable?

Should VaR be important to you? Consider this example before you answer. Suppose you are an insurer that specializes in hurricane insurance on Cape Cod. If a hurricane hits, you have losses on all your properties. In no hurricane, no losses. I think it is reasonable to assume that the probability of a hurricane is less than 1%. Thus the 99% VaR = 0. According to the VaR rule you need zero assets.

Now tell me, is our specialty insurer really providing insurance?

My answer to td's last question is Yes! We want to limit ourselves to coherent risk measures.