October 2012 Q3

[Simon James]

Hopefully, lots of candidates were able to calculate the capital, and hence the premiums for the 4 individual companies, using the gamma distribution with suitable parameters (looking up the parameters on the tables).

Interested to hear how you tackled the capital for the portfolio (and hence the diversification benefit). Did you try the same technique as above (and run into problems) or do something different?

 

[DevonMatthews]

We can just add the sample means and variances can't we? They were uncorrelated.. I remember calculating a fairly large diversification benefit.. (I made an approximation)

 

[14099]

Simon, I started out trying to fit a gamma distribution to the aggregate loss over the three companies, found that my alpha parameter was well outside the range of the tables at the back of the paper, so abandoned that and used a normal approximation instead. I didn't think it would be a good approximation given the skewness of operational loss distributions, but that was all that came to mind at the time.

I commented on the probable lousiness of the normal approximation, and how it would likely overstate the diversification benefit, in the next part of the question.

What did you think of Q2? The preamble was strange to say the least: a poor, undeveloped country, probably with just a fledgling insurance industry, trying to implement SII style regulation!

 

[ActPass]

For the portfolio capital calculation, I also fitted the Gamma distribution by adding up three sample means and variances (due to independence assumption), then calibrated the Alpha and Lambda. The Alpha was out of the range of the provided table, but I did some approximation by looking up the available table and scaled it up. The diversification benefit will then be (sum of the three stand-alone capitals -the portfolio capital). I know it can't be exact, but looking at some of the past examiner's reports, they would accept any reasonable numerical answers. Hope they thought mine was reasonable too

As for Q2, isn't the whole point that you can't simply apply Solvency II into this specific "small, poor and undeveloped country"? So you need to do something suitable to fit the country's situation. By the way, the question doesn't say to introduce Solvency II, but "to introduce similar risk-based regulation" via the Solvency II expert.

 

[bapan]

I am not sure we were required to add up the means and variances. Adding them would mean you are implicitly assuming that each business contributes to losses in equal proportions (unless you state that you are making this assumption in the first place).

I look at these being three independent portfolios.
The individual 1-in-200 capital numbers for each business were computed in part (ii). Use them. (Here no approximation is required as all values were in the tables)

The sum of these capital numbers gives the un-diversified capital number for the portfolio.

As the businesses are independent, we can compute diversified capital using square-root of the sum of squares of the 3 capital figures. The difference gives the quantum of diversification benefit for the overall portfolio. I got the diversification benefit of around £ 71K.

I thought using normal approximation to estimate capital at the tail would be incorrect as the tail of the portfolio is likely to be long and fat tailed whereas a normal distribution is nothing but.

Am I missing anything here?

 

[DevonMatthews]

I also recall a diversification benefit of around that much, but I didn't do it in that way. IF your going to use that method then you will have a correlation matrix thats just equal to the identity matrix.. so your diversified capital is just the square root of the sum of the squares.

I can't think of any reason why adding the individual numbers & finding the 99.5th percentile of the aggregate distribution is wrong.. Maybe they both arrive at the same answer.. I can't be bothered confirming this.

 

[bapan]

Devon

You are correct. My method is equivalent to using the identify matrix. This is okay to do as independence implies linear correlation = 0 as well.

I guess if you want to do it using aggregate mean and variance, you are effectively assuming that the aggregate distribution is gamma (the question does not tell you that explicitly but you would need to assume that).

I did some number crunching at home:
If we do it my way (as described in an earlier post): DivCap = 110,560
The UndivCap would be just sum of each component capital i.e. 181,442 ;
=> Div = 70,882

If we fit a gamma to the aggregate distribution: DivCap = 120,211 (note this number has been computed in Excel)
You have to assume UndivCap as sum of capital computed earlier i.e. 181,442
=> Div = 61,231
[But: You would invariably be making an 'incorrect' assumption that sum of gamma is gamma where the scale parameters of individual components are actually different. Note you have computed the individual capital numbers assuming gamma distribution]

If you want to avoid this assumption you would have to compute the un-diversified capital in some other way which I am not sure what it can be. Also, do we really need to re-do part (ii)!

PS: If we fit an approximate 'Normal' distribution to the aggregate distribution: DivCap = 104,031. This is much lower than any of the other numbers as Normal is a thin tailed distribution

 

[ActPass]

A few thoughts:

- The square root approach seems to be another way to go, and certainly has its own merits (eg quicker to calculate.. etc)

- However, isn't the square root approach also based on some implicit assumptions? I've done some research. The square root approach is based on two assumptions:
1. Each individual risk (i.e. each company in the portfolio in this case) has an elliptical distribution and that the joint distribution of the risks is multivariate elliptical.
2. There is a linear relationship between each risk factor and the resulting capital requirement.
I am not too sure if the question will guaranteer to meet these conditions.

- From a practical point of view, both aggregation gamma and square root approaches seem to produce the results in the same order of magnitude.

- For some reasons, rightly or wrongly, the wordings of the question appear to lead to aggregation gamma approach (obviously i was not the only one had the impression)

- Given the limitations, both approaches should probably given credit, particularly under exam conditions. ?

 

[bapan]

Hi

Yes, you are correct about both the properties.

I believe both are satisfied in this case

• If I remember from some work I did on elliptical distribution, Gamma does belong to that family
• There is linear relationship of the risks (it is zero correlation as they are independent)

The question is whether assuming aggregate distribution is also gamma is a valid approach or not (given it produces similar results) ! Well, statistically it can be proved that in this case the aggregate distribution will never be gamma albiet the shapes are similar. But again when are the different families of aggregate loss distribution look really different !!

I guess Simon may give some insights whether he thinks the examiners will allow both approaches?

 

[14099]

When I first read the post suggesting the root of the sum of squares approach, I was kicking myself for not having done this. I used a normal approximation for the aggregate distribution, which I thought would be a poor one at the time but I didn't see what else to do after having tried to fit a gamma distribution and finding the shape parameter to be well out of the range of the table.

However, looking at the question again they do ask you to calculate the aggregate capital requirement using a VaR methodology. I'm not sure doing the root of sum of squares counts as a VaR methodology, at least not explicitly.

It was probably worth only a couple of marks and I expect the exam report will state 'credit was given for any reasonable approach'. Whether they'll consider the normal approximation to be reasonable, given how skewed and fat-tailed the aggregate distribution will be, is debatable. I suppose you could have chosen any fat-tailed, positively skewed distribution and tried to fit that but it wouldn't have helped you unless you could evaluate the 99.5th percentile of it.

In the context of the whole exam I was more perturbed by Q2 than I was by this, especially the second half on whistleblowing. Ten marks on a subject barely mentioned in the course notes or in any of the other reading I did. My answers were just based on common sense, no particular subject knowledge.

 

[DevonMatthews]

That's a good point, the square root of the sum of the squares formula is not a var methodology, but it's inputs are 3 economic capital values calculated using var. The actual formula we refer to isn't actually ever stated in any official course reading.. It only ever appeared on one past exam.. And tHe solution contained little detail.. Not even the actual formula.

I only ever remember reading it in a paper I was reading on economic capital whilst studying for this subject..

 

[DevonMatthews]

Also, can't we not combine the 3 VARs to calculate economic capital directly since it isn't sub-additive?

I.e we need to generate a combined distribution THEN find the 99.5th percentile?

 

[bapan]

All

I stand myself corrected. I have managed to convince myself that the "root sum of squares" approach will not produce the correct VaR for the aggregate distribution in this case.

This is because of the skewness of the gamma distribution and hence the likely skewness of the aggregate distribution. [I was wrong to claim that gamma will be in elliptically contoured family]. In fact, the "root sum of squares" method systematically underestimates the capital even if the individual components are independent (and hence with correlation zero) in such cases.

Interestingly the 99.5% VaR for the aggregate distribution obtained using simulation (1m) is 123,280. This implies that we need to apply a correlation of +10% between three companies to "correct" for the skewness so that we can apply the "root sum of squares" formula.

This number is 'close' to that obtained using the approximation that aggregate distribution is gamma. Not sure if the numbers we are working with results in these two numbers to be so close given that sum of gamma can be equated to a gamma only when scale parameters are same ! [In this case they are not even close]

 

[Simon James]

Hi all

Thanks for an interesting discussion, which mirrored our own thoughts!

My immediate thought was to try to apply Approach II to the combined portfolio. Due to independence we can simply add the means and variances. Now, try to fit a Gamma - alpha = 5.68 and 1/lambda=8800. Unfortunately these aren't tabulated, so probably not the intended route.

(BTW, doing the maths in Excel gives capital of 120,211 - as Bapan correctly calculated below. However this still doesn't feel quite right as there's no reason why the combined portfolio should be a Gamma due to the different scales)

So, what to do? Look at the number of marks available: [6] for 4 parts to the question - there's probably only one mark (maybe 2?) for this calculation. There must be some approximation. The examiner's route was to use the square root of the sum of the squares. (Full marks to Bapan!)

Is this mathematically correct? Probably not. This approach is only an approximation using Normals, never mind Gammas. However it is a widely(?) used calculation (and there is a precedent in a previous exam). As Bapan points out, it does understate the "true" capital requirement.

Fortunately the Examiners are likely to award credit for any reasonable calculation (including making progress using Approach II before hitting the dead-end).

 

[DevonMatthews]

Requiring a value that isn't given in the tables doesn't stop you from answering the question..

Why can't you just fit a linear regression to the 2 highest values given in the table and estimate that way? When your that far into the tail the distribution is roughly linear anyway..

 

[markf268]

Problem with doing that is that it was a question worth 2 marks.

I started doing the sum of gammas to make a gamma but like everyone else who did ran into the brick wall so just assumed an answer as I could see where it was going so that I could get the remaining marks.

 

[Simon James]

Problem with doing that is that it was a question worth 2 marks.

I started doing the sum of gammas to make a gamma but like everyone else who did ran into the brick wall so just assumed an answer as I could see where it was going so that I could get the remaining marks.

Good exam tekkers!