Ch 7 - Lots of questions...

Hi all

I'm struggling through Ch 7. I'm struggling a little to see how much detail to go into and where to just accept the statements made.

Some particular questions that I have are given below. Sorry it's quite a long list. 6-8 are the ones I'm most concerned/confused about.

1. What are the benefits of capital allocation for reinsurance planning (from SA qu 7.2)?

Is it that it allows us to see where reinsurance is most effective by allowing us to see how the capital for diff LoB are impacted by diff programmes?

2. What are the benefits of capital allocation for investment strategy (from SA qu 7.2).

Is it that it allows us to see where asset allocations are most effective by allowing us to see how the capital for diff LoB are impacted by diff allocations?

3. Why is linear homogeneity a desirable risk characteristic given it ignores the diversification benefit for larger lines of business?

I have seen these characteristics given for asset portfolios (where this axiom makes sense) but don't understand why this one applies to GI liability portfolio.

4. p. 15 EPD def'n

"This is a variant of the above...set as the company's surplus capital" should this be "available capital"? Otherwise the non core reading text below is inconsistent. It seems to me it should be available as beyond this there is a default risk for the policyholder.

5. Mean loss transformed risk measure p. 16

Transformed loss differenced from the original mean loss to give the risk measure. Would this not be negative?

6. It's not immediately obvious to me why the proportional method violates diversification benefit principle.

Is it because there is no guarantee that a low correlation portfolio gets a lower charge this way than if it were considered separately.

Are we able to construct examples under this method where the principle is violated?

Is there a symmetry between this and the marginal method where uncorrelated portfolios are overcharged and therefore violate the diversification principle.

7. p. 24

“The game theory allocation is additive and coherent but only under additional conditions that prevent the risk measure recognising diversification effects.”

My understanding was that to be coherent it needs to recognise diversification? Both the risk measure and the allocation? Do the mean in this case the allocations just add so there is no actual diversification?

I'm just confused about this statement in general.

8. p. 25 Equalise relative risk

A-> If R(X) is the chosen risk measure and E(X) an exposure measure, we allocate capital such that R(X)/E(X) is the same for all LoB.


B-> Allocate capital until the selected risk measure is equalised relative to the expected losses across the whole business.

I understand A not B. I don't see how they are the same. Is A a meaningful def'n here where the risk measure is not the EPD?

Does B make more sense if the word expected is removed.

I'm just not clear what the method is here.

9. Co measures

I understand Co-VaR, Co-TVaR and Co-XTVaR.

Should covariance be like the more conventional def'n of covariance

E[(X – mu_i) (X – mu)]?

I'm don't understand how the Co-EPD would be calculated, or the reasoning behind its formulation. Any ideas on how it works in practice.

 

Yes. For example, to see how much reinsurance we might need for each class.

Yes. For example, if we know how much capital we have for each class then we can match more effectively.

The idea is that the multiplier is just a constant (eg could be related to currency, so that the risk measure is independent of the currency in which the risk is measured). Doubling the exposure to a risk therefore requires double the capital, which makes sense because doubling the position provides no diversification. If we're talking about a bigger portfolio of risks, however, then it does fall down because it doesn't allow for diversification (which is where the sub-additivity property makes more sense).

I would interpret available capital to be the same thing as surplus capital here.

The term "differenced from" doesn't define a particular direction for the subtraction. You would just use the positive value.

It's because no account is taken of the correlations between classes ie no allowance for diversification.

Assume you have 3 classes A, B and C that are all the same size according to the risk statistic, and that A and B are correlated with each other but neither is correlated with C.
If total capital = £90m then the proportional method would allocate £30m to each as they are all considered to be equally risky.
If you wanted to allow for correlations, however, you might split the £90m total capital differently, eg £35m to A, £35m to B and £20m to C. Here, C would hold less capital than the others as it represents a degree of diversification for the company.

Whereas the proportional method makes no allowance for correlations, the marginal method gives too much.

The allocation method has to recognise diversification in order to be coherent. Shapley game theory does do this. However, the conditions for the risk measure to be coherent do not mention diversification. Indeed, it would be undesirable in this case as we would effectively end up double-counting the diversification effect.

It doesn’t really matter what the risk measure is – EPD is just one example. The selected risk measure is R(X) and the risk premium E(X) is our best guess as to the expected losses. So, to equalise the risk measure relative to the expected losses, we need to have a constant R(X)/E(X) for all risks.

You need the word “expected” because, at the time when you need to set the capital to cover the losses, you can’t know for sure how big your future losses will be.

There are lots of papers around that hint at this sort of stuff, but it's very mathematical. You could try Google searches if you're interested in finding out more. It's worth bearing in mind that the SA3 examiners don't usually concentrate on 'hard maths' like the stuff in this chapter so it might not be worth spending too much time on it unless you're really interested or have lots of spare time.

Coralie

 

Proportional method violates diversification

6. It's not immediately obvious to me why the proportional method violates diversification benefit principle.
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I agree with the question, in fact I'm convinced that the notes are wrong here and that the proportional method does not violate the diversification benefit principle.

The "diversification benefit criteria" is:
"The allocation for a sub-portfolio (or coalition of sub-portfolios) should be no greater than if it were considered separately."

Proof: Let X, Y be our sub portfolios and R() be our risk measure then we have:
1.)The risk measure is coherent so R(X+Y) < R(X)+R(Y) is a known fact.
2.)Proportional method's allocated amount to risk X is then R(X+Y)*{R(X)/(R(X)+R(Y))}
3.)The capital allocated to X if considered separately is R(X)
4.) The question is then just are we sure that:
R(X) > R(X+Y)* {R(X)/(R(X)+R(Y))}
but dividing by R(X) and multiply by R(X)+R(Y) we see that this is the same as R(X+Y) < R(X)+R(Y) which is known to be true from point 1.) above.

So we can see that "proportional allocation" + "any coherent risk measure" will never fail the "diversification benefit criteria"

I hope someone can confirm my thoughts on this or show me where I'm going wrong?

 

Hang on a second!

I’m concerned that there’s potentially a lot of misinformation flying around here and I’m afraid I consider the Profession guilty of propagating a lot of it, as I’ve made clear in these forums on several previous occasions.

Regarding Leela’s question 6, I can answer this now, but I’ll wait until I’ve got my copy of the course notes in front of me before I look at the others as I think I can give more helpful answers when I can see these in context.

Coralie tells us that the proportional method violates the diversification benefit principle “because no account is taken of the correlations between classes ie no allowance for diversification.” However, as doubleteam points out, the diversification benefit principle does not explicitly require the method to take account of correlations. It simply states that the allocation for each line of business must be less that if the LoB was considered separately. Coralie’s example does not address the question of whether this is true for the proportional method.

Leela reasonably asks, “Are we able to construct examples under this method where the principle is violated?” The answer is yes, although you may consider that the example I’m about to give you is a tad pathological.

Consider a problem in which one LoB has a negative allocation when considered separately. As doubleteam correctly argues, the fact that we require coherent risk measures to be sub-additive means that the magnitude of the allocations gets scaled down when we consider them as part of the overall portfolio. But if you scale down a negative allocation, you end up with an allocation that is greater than the one you started with, hence violating the diversification benefit principle.

So doubleteam, the short answer to your question is that, when you say “dividing by R(X) and multiply …”, if R(X) were negative then the inequality sign would switch at this point and you wouldn’t reach the same conclusion.

This may strike you as a bit of a pure mathematician’s counterexample and not a particularly practical approach to the problem. The important thing here, which is not really explained in the Core Reading, is that coherence is a highly mathematical concept and it is up to the practitioner to decide whether it is a desirable property is any given situation. A slightly more obvious justification for claiming that the proportional method is not coherent is to point out that it fails the “Riskless Allocation” principle. I can provide an example of this if needed but I think this claim is reasonably intuitive.

In conclusion therefore, the Core Reading is correct in its assertion that the proportional allocation method is not coherent as it does not in general satisfy the diversification benefit principle. An explanation would have been helpful though. I suspect the reason none was forthcoming is because the Core Reading author doesn’t know why it’s true either and was just cribbing from some paper. When it comes to capital allocation, the general rule seems to be: if in doubt, make it up! If you ask questions, you’re being a lot more conscientious than the some people who should be setting a good example.

 

Leela's q.7

The Shapley method satisfies the “Symmetry” and “Riskless Allocation” properties as a direct consequence of its definition. The difficulty is in establishing whether or not it satisfies the “Diversification Benefit” property. To my knowledge, no-one has yet been able to prove that coherence of the risk measure is sufficient to give coherence of the Shapley method, although I haven’t seen a counterexample to this either. Instead, it has been shown that the Shapley method can give coherent allocations when tighter restrictions are placed on the risk measure. Now you mention that you believe the risk measure should recognise diversification. Presumably you are referring to the requirement that \[\rho(X+Y)\leq \rho(X)+\rho(Y)\] However, notice the nature of the inequality sign here. It’s not a strict inequality. In fact a risk measure could satisfy \[\rho(X+Y)=\rho(X)+\rho(Y)\] and still be coherent. (The mean is actually a coherent risk measure, albeit not a particularly useful one!) So one way of obtaining coherent allocations from the Shapley method is to use a risk measure that satisfies this latter equation. However, such a risk measure would not really recognise diversification benefits in any intuitive sense. This is what the Core Reading means when it says that the Shapley method is coherent "under additional conditions that prevent the risk measure recognising diversification effects."

Incidentally, it makes no sense to speak of "double-counting" diversification benefits in this context. Essentially, the risk measure recognises diversification benefits through the "Sub-additive" property. The allocation method allocates the total risk measure to sub-portfolios. Equivalently, it can be seen as allocating the diversification benefit recognised by the risk measure to the sub-portfolios (so the allocation for any sub-portfolio is equal to the risk measure when considered separately + its allocation of the overal diversification benefit.) The important thing for it to satisfy the "Diversification Benefit" property is that it should not allocate a negative diversification benefit to any sub-portfolio.

By the way, I agree with Coralie that most of this is way beyond what's likely to be tested in the exam. So is a lot of what's in the SA3 notes. It's weird that they should include all this stuff.

 

Leela's q.8

On the contrary, it really does matter! For this to work, you need a risk measure whose value depends on the allocated capital. Otherwise, how are you supposed to influence the value of R(X)/E(X) by changing the allocated capital? You'd be stuck with one set of values for R(X)/E(X) and if they weren't already equalised across LoBs there would be nothing you could do about it. So for example EPD can be used but VaR can't.

For anyone interested in Co-EPD and co-measures in general, the logic behind them is detailed in the following paper:

Kreps, R. (2005) Riskiness Leverage Models https://www.casact.org/library/05pcas/kreps.pdf

 

RE: Hang on a second!

Thanks td290 for a very thorough answer to my question. Indeed as you pointed out I was assuming that risk measures would always be positive, but looking more closely at the requirements of a coherent risk measure I see that the "Translational invariance" rule requires risk measures to give negative capital to a risk free portfolio. This then sinks any hopes of the proportional allocation method being coherent.

My next question to you is then regarding the XT-VAR formula given in the notes. Considering the yth percentile for the T-VAR and XT-VAR calculations, the notes state:

XT-VAR(x) = T-VAR(x) - E(x)*y%

But it looks to me like the formula should be:

XT-VAR(x) = T-VAR(x) - E(x)

Furthermore the notes go on to say XT-VAR is a coherent measure but it looks to me that because it subtracts the mean it fails the monotonicity criteria.
Consider a portfolio with a massive expected profit and small spread compared to another portfolio with a massive guaranteed loss.

 

Yes - you're not the first to point this out! In fact this was something I raised a while ago (see http://www.acted.co.uk/forums/showthread.php?t=6855) and has since been corrected (http://www.acted.co.uk/Docs/2013/Corrections/SA3 Corrections 2013.pdf).

 

Chapter 7 - Capital Allocation

Hi,

I still could not fully understand why the proportional method is not a coherent allocation method because of the violation of "diversification benefit" principle.

The proportional method scales the resultant allocation to provide an internally consistent allocation of diversification benefits such that the sum of individual allocations is equal to the whole.

so P(X+Y) = P(X) + P(Y) .............(1)

A capital allocation method is in line with the diversification benefits principle if the allocation for a sub-portfolio (or coalition of sub-portfolios) is no greater than if it were considered separately.

so P(X+Y) <= P(X) + P(Y) .............(2)

If this is the case than equation 1 meets the requirement of equation 2.

Why then is the "diversification benefit" principle violated under the proportional allocation method?


Thank you. Would appreciate any assistance in regards to this.

 

I think the problem is described on page 22, where it says that "proportional methods neither penalise highly correlated sub-portfolios nor give credit for low correlation sub-portfolios."

So it's allocating capital inaccurately, without allowing properly for where the diversification / correlations arise.

So if one class is allocated too little capital, it follows that another class must be allocated too much capital.

So the allocation for a class may be higher than if it were considered separately, which is against the diversification benefit principle.

 

MorningtonCrescent, I would have thought that your reasoning implies that any allocation method which takes diversification into account violates, paradoxically, the diversification benefit principle.

Take Coralie's original example:

In the example above, if we use a method that does allow for correlations, we end up allocating more to A and B than if they were considered separately?

I've read through a number of past posts around chapter 7 but haven't found this question mentioned - apologies if I missed it, or if there's something obvious I'm missing!

EDIT: ok, so the obvious thing I missed is that the allocation doesn't need to total 90 - it could for example be [30,30,20] for [A,B,C]