Methods for risk aggregation

Can you confirm that there are two methods that are used for aggregation of risks inpractice?

1. by applying copulas

2. by applying a correlation matrix

 

Yep,

To me looks like the list is this exhaustive!

 

Thanks Edwin.....how would you decide which method to use? Any pros or cons for these approaches? How is done in the industry?

 

Method 3:
Office politics and top management pressure

 

Hi V,

I work in capital management and the decision of which one to use involves;-

1) Going for copulas if you believe some of your risks are non - normal e.g skewed (it is generally known that the underlying distributions of the relevant risks of an insurance or reinsurance undertaking are not normal distributions. They are usually skewed and some of them are truncated by reinsurance or hedging effects).....

.....and the relationships between your risks is not linear for example if you can justify tail dependencies. Remember correlation matrix is linear dependence > is pearson's rho > implies you are making an assumption that your risks are normal distributed and that the relationship between them is jointly elliptical.

2) If you think you can explain Copulas to CEO (see Oxymoron's post below)

 

I semi-agree with some of the other posters in this thread but would like to clear up a couple of misconceptions.

When you apply a correlation matrix you are effectively applying a copula. The copula intuitively provides the information about the joint behaviour of random variables. The only difference is that you are applying the copula corresponding to a multivariate normal distribution and because this is such a common thing to do it has somehow elevated the "applying a correlation matrix" approach to be thought of as something else. It isn't.

Part of Edwin's post

"Going for copulas if you believe some of your risks are non - normal e.g skewed"

seemed to imply that if you use a correlation matrix then all of your marginal risk distributions will be normally distributed. This is not the case. When you apply a correlation matrix, you are changing the joint behaviour of your random variables, not their marginal behaviour. You can very easily generate two pareto random variables which have a given rank correlation as specified.

I would also like to pick on another of Edwin's quotes:

"If you think you can explain Copulas to CEO"

This may have been said in jest. A CEO should not be expected to understand the copulas used within the Capital Model. They should, however, understand what tail risk is and that the Capital Model will reflect that when things go bad for one part of the business, it is likely that they'll go badly for another part. CEOs can understand joint exceedence probabilities, and really that should be all they need to know.

 

Hm, that is an interesting view....I never thought of a correlation matrix as a type of copula. It is nowhere mentioned in the ST9 materials, I believe.

Can it be stated in the exam?

 

Point 1, there is no Copula called correlation matrix. You are trying to say that they do the same thing, otherwise they are different things;- one is calibrated by linear dependence, the other by non-linear dependence.

Point 2;- (strong)
Whenever you use a correlation matrix “you are applying the copula corresponding to a multivariate normal distribution” (posted by you)
A strict requirement is that a multivariate normal distribution is made up of normal marginals – (my point; you can’t use pareto)!

Now a correlation matrix is pearson’s rho hence you can never use rank correlation. See example below;-

[3 5]*[1 0.25]*[3]
[0.25 1] [5]

= sum of squares, this is just;-you can test 3^2+5^2+2*p*3*5. Use p say 0.25. The p in x^2+y^2+2*p*xy is strictly pearson’s rho. (My point; you can’t use rank correlation)

P.S;-
Kendall’s Tau, Spearman’s rho are used in the calibration of a COPULA to define a dependence structure.

Point 3;- The comment is not as digital as you make it seem.
Point 4;- I don’t get the hello, was it originally posted by me?


((Let me know If I should expand...))

 

Not about CEOs understanding Copulas or about the Copulas themselves.

There is a huge conservatism bias when it comes to setting reserve numbers. The first question asked when you present the numbers to the executives is "It was only X last year, why is it so high this year?" or "Why have the reserve numbers fallen so much since last year - won't the regulator notice that?". It's always in reference to what was set last year.

Then there's the fight between risk and sales - sales wants to keep reserve as low as possible while risk and legal want to keep it as high as possible. Modeling too much of the tail risk will make sales a sad panda and will mostly not be allowed anyway.

Copulas provide a diversification advantage till a particular point and then it turns to disadvantage - which will also be questioned.

The politics of setting reserves should not be underestimated.

 

I'm struggling to find anything in the notes or textbooks about using a correlation matrix approach, which is somewhat surprising given its used in (most?) Individual Capital Assements and the Standard Formula for Solvency II. Nevertheless I guess it means it's not on the syllabus.

I wouldn't say I'm an expert on this, but I am familiar with it from my previous job. I think Shillington is correct - from what I remember, using a correlation matrix approach implies that the joint distribution is normal and hence the copula is also normal.

When you use a correlation matrix approach you would calculate each of the univariate VaRs first at the desired probabilty level, giving a vector V of univariate VaRs. If the correlation matrix is C, the aggregated VaR would be sqrt(transpose(V) x A x V).

In practice the univariate VaRs could be calculated by any means using any distribution. However, unless you're using normal distributions it would be inconsistent with your aggregation approach. So you'd be making an implicit approximation by using a correlation approach. One way to adjust for this is to model a joint scenario in addition to the correlation result. However it can be difficult to calibrate/specify the joint scenario to model.

In the notes, it mentions that normal mixture multivariate distributions can have elliptical PDFs, so this may also be consistent with using a correlation matrix approach. But it's not something I've heard elsewhere.

Hope that helps a little.

 

Hi, it is not clear what part of Shillington's comments you agree with, but here are my points;-

1. Do not use the word copula in reference to a correlation matirx, one is based on linear dependence the other on non-linear dependence, rather say they do the same thing
2. Shilligton is wrong in saying you can use a Pareto distribution in a correlation matrix - which you seem to get in your explanation of allowing VAR computed in a different sense
3. Shilligton is also wrong in saying you can use Kendal's tau with a correlation matrix
4. To understand points 2 and 3 above, note that this multivariate normal that we refer to as defining the correlation matrix is defined in terms of spearman's rho, it's mean and variance (Probability 101).

 

Edwin, sorry for the late reply. I did reply a while ago but the forum decided to post a blank post... irritating.

First off I would like to say that I completely forgot about the "correlation matrix" approach to risk aggregation, where you multiply vectors of risks with the correlation matrix. I interpreted it as using a Multivariate Gaussian copula with the selected correlation matrix. (Who actually uses the former in practice?!?!?)

Now, these are actually the same thing for normal random variables (so long as you switch between pearsons rho and spearmans rank). As you correctly point out you cannot use a correlation matrix to aggregate pareto random variables and get the correlation structure (you can when you use a multivariate gaussian copula which is what I was referring to). I can, however, very facetiously point out that you can use a correlation matrix to aggregate pareto random variables, the answer you get is an aggregation, it just doesn't have any natural interpretation (but you try and give me a natural interpretation of what 50% correlation means!).

The point I was originally trying to impress was that when people talk about copulas I often feel they mean "non-normal" copulas and forget that you can apply a multivariate normal copula to risks in just the same way you would apply a gumbel copula etc. That was the point I was trying to make.

 

Cheers Shilling ton....50% when I aggregate using a correlation matrix based on rho (pearson's) means linear dependence and therefore says if one risk goes up by x, the corresponding risk will go up by 50%x. That's the natural interpretation when you use the right things, I agree with you though....there is no natural interpretation for using "pareto random variables" in a correlation matrix!

sure Shillington, maybe you should study ST9...I have a feeling you will do very well given your practical approach and insights, I failed first time though FA!