Axioms of coherence

[Alastair_in_SA]

Hi there

I think the translation invariance axiom may be wrong. The ActEd notes say:

F(L + k) = F(L) + k

However, Wikipedia and some of my lecture slides from CT8 say it should be:

F(L + k) = F(L) - k.

That is, the case where k is a risk free asset, adding it to the portfolio L reduces the risk by k.

Any clarity here would be great.

Thanks
Alastair

 

[Edwin]

Hi there

I think the translation invariance axiom may be wrong. The ActEd notes say:

F(L + k) = F(L) + k

However, Wikipedia and some of my lecture slides from CT8 say it should be:

F(L + k) = F(L) - k.

That is, the case where k is a risk free asset, adding it to the portfolio L reduces the risk by k.

Any clarity here would be great.

Thanks
Alastair

Hi Alastair, You have a company with two business lines, A and B say the retail branch of a bank and a Life insurance division (common in South Africa) . You want to find the total amount of some say Market Risk capital. You go to the branch and model the loss distribution as L, then go to the Life Insurance company to try and model the loss distribution. By the time you get there the CRO gives you their risk capital number, he says "we have quantified our Market risk to be k".

Then you go back to your desk and have to answer your initial question, "what it the total amount of Market risk capital?"

You have your loss distribution from the branch L, and you decide to use Var to find the capital (for all business lines A + B) i.e F(L+k), then you remember you have the capital for the Life Insurance business line.

So u start worrying with F(L) knowing that you have k, hence F(L)+k.

P.S;-
- I have a technical mathematical proof I can post on request if that is what you are looking for.
- Let us extend the solution;-

Suppose now that the Life insurance company gave you their own loss distribution Y. You now want F(L+Y) and here follows sub-additivity. Assuming there is correlation between Market risk at the bank and at the Life division of your bank.

(I was typing this fast, feel free to ask for clarity if not clear, also you can modify example such that Y is a different risk within the same life division and hence k the capital of that risk)

 

[Alastair_in_SA]

Thanks.

Yes please send me the proof. Here is the link to Artzner et al. (1999) in which he sets out the axioms: http://personal.fmipa.itb.ac.id/khreshna/files/2011/02/artzner1999.pdf. Translation invariance is on the bottom of page 208 and top of page 209. Perhaps the results are the same but Artzner et al. have defined F (or rho in their notation) to mean something slightly different to what ActEd have defined it to be.

 

[Edwin]

Thanks.

Yes please send me the proof. Here is the link to Artzner et al. (1999) in which he sets out the axioms: http://personal.fmipa.itb.ac.id/khreshna/files/2011/02/artzner1999.pdf. Translation invariance is on the bottom of page 208 and top of page 209. Perhaps the results are the same but Artzner et al. have defined F (or rho in their notation) to mean something slightly different to what ActEd have defined it to be.

PROOF;-

Let L be a random variable and k be a real number.

VAR(L+k) = inf{x|P(L+k<=x)>=a}

=inf{x|P(L<=x-k)>=a}
Let s = x-k
=inf{s+k|P(L<=s)>=a}
=inf{s|P(L<=s)>=a}+inf{k|P(L<=s)>=a}
=inf{s|P(L<=s)>=a}+k
=VarL+k

PS;- but please do try to read my example above.

Counteracting your example;- See remark 2.10 on page 210 of your paper. It is assumed that F(k*-r)=k hence;-

F(L+k*r)=F(L)+F(K*r)=F(L)+-k....I only looked at the math in your paper, didn't try to understand, but that's what's happening.

Let me know if you are happy.

 

[Alastair_in_SA]

Thanks.

Yes I did read your example, it makes sense.

Thanks for your explanation of this axiom using VaR as an example. I was expecting something more general, but then I remembered we are talking about an axiom, which obviously can't be proved. So your example was good, however I don't remember real analysis well enough to justify the step where you split out the sum in the infimum - if it's simple can you explain that in words?

I am just not convinced (yet) that the axioms given in ActEd / Sweeting are saying the same thing as Artzner et al. (just in case you don't know Artzner et al. put forward these axioms in the first place). Your example in the first reply makes sense, and your VaR example in the last reply (assuming the splitting of the sum in the infimum is allowable) makes sense, but what the original axiom says and what the ActEd notes say are different. Artzner et al. says that adding an initial amount k to the portfolio and investing it in the reference instrument decreases the risk measure by k. ActEd says adding k to the initial portfolio increases your risk measure by k.

 

[Edwin]

Thanks.

Yes I did read your example, it makes sense.

Thanks for your explanation of this axiom using VaR as an example. I was expecting something more general, but then I remembered we are talking about an axiom, which obviously can't be proved. So your example was good, however I don't remember real analysis well enough to justify the step where you split out the sum in the infimum - if it's simple can you explain that in words?

The English version is exactly in my example above.

I am just not convinced (yet) that the axioms given in ActEd / Sweeting are saying the same thing as Artzner et al. (just in case you don't know Artzner et al. put forward these axioms in the first place). Your example in the first reply makes sense, and your VaR example in the last reply (assuming the splitting of the sum in the infimum is allowable) makes sense, but what the original axiom says and what the ActEd notes say are different. Artzner et al. says that adding an initial amount k to the portfolio and investing it in the reference instrument decreases the risk measure by k. ActEd says adding k to the initial portfolio increases your risk measure by k.

1) Did you see this;-

Counteracting your example;- See remark 2.10 on page 210 of your paper. It is assumed that F(k*-r)=k hence;-

F(L+k*r)=F(L)+F(K*r)=F(L)+-k....I only looked at the math in your paper, didn't try to understand, but that's what's happening.

2) Maybe you must understand what Artzner portfolio this is, if you have R4000 invested in the stock market and put k = R500 more, it decreases your risk?

Please read the whole of the Artzner paper;-

1) Note that ST9/Sweeting is talking about F(L+k) while Artzner is talking about F(L+K*r), r is the total rate of return and Remark 2.2 on page 207 says that F(r)=-1.

 

[Simon James]

I think we are talking about the same concept, but using different terminology.

The idea is that if a constant amount is added to the losses the risk measure should go up by the same quantity.

Similarly, if a constant amount is deducted from the losses the risk measure should go down by the same quantity.

Let us say that we have loss L, risk measure F and hold capital C = F(L). Then the adjusted loss position is L - C, so F(L - C) = F(L) - C = C - C = 0 as we might expect (ie we need no further capital).

In the Wikipedia case of a risk-free asset being added the portfolio, the return on that asset can be considered a reduction from the loss and hence from the risk measure.