Chapter 15

[Cheng]

Hi! I have some questions from Chapter 15..

1) in deriving equation 2.3, there are some mathematical workings involving ^ and v. Though the workings make sense, I'm not sure if I'm able to come up with something like this in the exam. So I'm wondering if we're expected to understand/ find out more about the mathematical workings involving ^ and v?

2) Eqn 2.3 makes sense where the layer loss cost = ground up loss * differences in the exposure curve with relativities (L+D)/M and D/M. However, for equation 2.4, I don't understand why you need to devide by [1-G(d/(M+d))]? Is there any intuitive explanation behind eqn 2.4?

3) also in page 20, the notes say 'the presence of original deductibles will increase the reinsurance rates', it means that reinsurance will be more expansive where there's original deductibles? I don't see why is that the case, although the insurer is not liable for the original deductible, so is the reinsurer and why does that make reinsurance more expansive?

4) in page 21 under the heading of Treatment of stacked limits, the notes say 'if the layers are stacked then the underlying contract is essentially 1.4m xs 100k and the reinsurance covers the range [600k, 1.1m]. If they are independent, the reinsurance only affects the 1m xs 500k contract, covering the range [1m,1.5m]'

how is the range [600k,1.1m] obtained? Also when the contracts are independent, shouldn't the reinsurance cover the range [0.5m,1.5m] instead of [1m,1.5m]?

5) under Catastrophe XL rating, what does it mean by 'long return periods' and why does it make curves difficult to estimate?

6) eqn 6.2 says C(delta I) = E(N)E[X^(delta I +D) - X^D]
= E(N)[S(D)delta I]
I don't see how to get the second line?

Thanks in advance!

 

[Katherine Young]

1) in deriving equation 2.3, there are some mathematical workings involving ^ and v. Though the workings make sense, I'm not sure if I'm able to come up with something like this in the exam. So I'm wondering if we're expected to understand/ find out more about the mathematical workings involving ^ and v?

Questions on this topic can and do come up. You should be prepared to use these results in the exam, and be able to understand / interpret them, should the examiners ask you to do so. The examiners can also ask you to prove a result like (2.3), as well as any other proofs given in the Core Reading. Personally, I think it’d be a little mean if they ask you to do anything more unusual.

2) Eqn 2.3 makes sense where the layer loss cost = ground up loss * differences in the exposure curve with relativities (L+D)/M and D/M. However, for equation 2.4, I don't understand why you need to devide by [1-G(d/(M+d))]? Is there any intuitive explanation behind eqn 2.4?

(2.3) gives CL, which is an expectation. Recall that a conditional expectation is E(X I X>d) = E(X)/P(X>d). Therefore, (2.4 ) is giving us a conditional expectation, because we need claims to first exceed the deductible, before we think about the loss to the insurer.

3) also in page 20, the notes say 'the presence of original deductibles will increase the reinsurance rates', it means that reinsurance will be more expansive where there's original deductibles? I don't see why is that the case, although the insurer is not liable for the original deductible, so is the reinsurer and why does that make reinsurance more expansive?

Neither the insurer nor the reinsurer is liable for the original deductible, which is borne by the insured.

If a deductible is introduced, this could have a big impact on the direct writer (plenty of claims won’t breach the deductible, so the insurer won’t see these claims at all anymore).

However, the reinsurer’s claims won’t drop by nearly so much, because it would never see all of those small claims, whether or not the deductible is in place.

So, relatively speaking, a direct writer is more affected by a deductible than a reinsurer.

Let’s take an extreme (albeit unrealistic) example; let’s say that the direct writer’s expected claims reduce by a lot, but the reinsurer’s expected claims don’t reduce at all. Therefore the direct writer can reduce its premium, but the reinsurer won’t reduce its rate at all.

So the reinsurance rate (expressed as a percentage of the direct writer’s premium) will increase.

4) in page 21 under the heading of Treatment of stacked limits, the notes say 'if the layers are stacked then the underlying contract is essentially 1.4m xs 100k and the reinsurance covers the range [600k, 1.1m]. If they are independent, the reinsurance only affects the 1m xs 500k contract, covering the range [1m,1.5m]'

how is the range [600k,1.1m] obtained?

The first 100k of a (ground up) loss won’t impact the cedant (because its lowest layer is 400k XS 100k). It then has to pay the next 500k, until the reinsurance kicks in. So this is a total of 600k (ie 100k + 500k) from the ground up. The RI cover is 500k XS 500k, which, in terms of the ground up loss, is on the range [600k, 1.1m].

Also when the contracts are independent, shouldn't the reinsurance cover the range [0.5m,1.5m] instead of [1m,1.5m]?

Similarly, the first 500k of a ground up loss won’t impact the cedant’s 1m XS 500k layer. The cedant then pays the next 500k (taking the ground up loss to 1m), before the RI kicks in. After this, the RI covers 500 XS the first 1m (of ground up loss).

5) under Catastrophe XL rating, what does it mean by 'long return periods' and why does it make curves difficult to estimate?

A return period (of n years say), is equivalent to saying that a catastrophe occurs once in every n years (250 years say).

So a long return period is like saying that catastrophes don’t happen very often at all.

Rating curves are difficult to estimate for Cat XL because we simply don’t have relevant historic data available to estimate a curve.

6) eqn 6.2 says C(delta I) = E(N)E[X^(delta I +D) - X^D]
= E(N)[S(D)delta I]
I don't see how to get the second line?

The first line basically says that C(delta l) is the expected number of claims multiplied the expected amount of a claim within the narrow layer. I think it’s this “second bit” (the expected amount of a claim within a narrow layer) that’s bothering you.

For a very narrow layer, we can approximate this “second bit” by just saying that it’s approximately: (the probability of hitting the layer, multiplied by the height of the layer).

Well the probability of hitting the layer is just S(D), and the height of the layer is just dl.

 

[mario]

Please can someone go through the steps of deriving equation (2.4)? The specific part I'm having trouble with is following Katherine's definition of conditional probability, my denominator is P(X > d) = S_Y(d/d+M).

But I can't convert the survival probability in terms of the exposure curve function G.

I've tried using the derivative of the LEV is the survivor function and that the integral of the survivor function is the expectation, but not really getting anywhere.

Thanks

 

[lwang]

(2.3) gives CL, which is an expectation. Recall that a conditional expectation is E(X I X>d) = E(X)/P(X>d). Therefore, (2.4 ) is giving us a conditional expectation, because we need claims to first exceed the deductible, before we think about the loss to the insurer.

Hi Katherine,

I have the same question as mario. I cannot see how to use conditional expectation E(X|X>d) = E(X)/P(X>d) to derive (2.4).

Thanks for help!

 

[MorningtonCrescent]

If the exposure curve is based on ground-up data, then a loss has to be greater than (d+M) in order to hit the layer.

So replace all the (M) in equation 2.3 with (d+M).

However, since we're trying to find a conditional expectation (as Katherine says, "we need claims to first exceed the deductible, before we think about the loss to the insurer"), we need to divide by P(X>d). Well, the analogous denominator for loss curves is (1-G(d/(d+M)).

 

[mario]

Well, the analogous denominator for loss curves is (1-G(d/(d+M)).

Thank you for your explanation... I understand the logic behind the calculation, but still can't actually manipulate P(X>d) to actually get to the required answer in terms of G.

That is, when I try to convert P(X>d) in terms of the survivor function, and then in terms of the exposure curve function G, I can't get to the right answer.

Please can someone post the derivation of P(X>d) = (1-G(d/(d+M)) step by step? Thanks

 

[indexo]

Can someone please help? How did we derive the denominator for (2.4)? Is it required to know the step by step?

 

[Katherine Young]

Hi Indexo,

I've knocked up an answer for you, hopefully of some use. See the file attached.

In fact I suspect your question is probably going a little further than the examiners are interested in. So this explanation is not a rigorous proof. Nevertheless, if you were asked to derive equation (2.4) in the exam I’m sure this answer would score pretty well.

Best wishes,

Katherine.

 

[Raiyan]

Hi,

Would you mind re-uploading the file as it doesn't seem to work?

Many thanks,

 

[Darren Michaels]

Here you go