Original Loss Curves

Hello there, I have quite a few questions on Chapter 15, would greatly appreciate any help!

1. From eqn 2.1, it is stated that G(1) = 1. Intuitively this is clear, but just wondering if there's a mathematical derivation to this from eqn 2.1?

2. In Section 4.2, we saw the ISO's approach to derive ILFs (pg 29) using closed claims. However, it was mentioned in pg 28 that it is not appropriate to ignore open claims. Does the ISO's approach take this into account somehow?

3. In the last paragraph of Section 5.2, the core reading says that "ILFs derived from Riebesell curves are scale invariant and do not need to be adjusted for inflation or changes in currency. (provided the attachment points remain sufficiently high for the curves to be valid).". However, I do not understand how the high attachment points relates to scale invariance?

4. Is it right to think that Riebesell curves are just specific ILFs that only consider multiples of 2 of the sum insured, while general ILFs can include any limit?

Section 6.1: Inferring treaty loss distributions from exposure rating
5. I am struggling to understand what this section is all about.
- Is the 'exposure rating' here just referring to exposure curves specifically?
- Is the section trying to start from an exposure curve obtained from a treaty without any aggregate features, infer the underlying treaty's aggregate loss distribution; then only estimate the impact of the aggregate features on the treaty and adjust the exposure curve rates from there?

6. Here, we are trying to obtain the aggregate loss distribution. Previously in pg 7, it was stated that we only look at the limited expected severity and ignore frequency as we are assuming frequency and severity to be independent.
- Is it right to say that in pg 7, we are just using the severity to obtain our loss costs at each limit, hence frequency is not required; while in Section 6.1, both frequency and severity distributions are required as we need to obtain the aggregate distribution to price in the aggregate features?

7. In pg 35, "We can use the following approach to make sure that the severity distribution chosen is consistent with the exposure rating (and hence the underlying risk profile and assumed original loss severity distributions)."
- From what I understand, we are looking at a single layer (i.e. from attachment point of the treaty to the exit point), and we want to obtain a severity distribution for this layer, making it consistent with the original loss severity distribution that was used in obtaining the layer loss cost C_L for this layer (under the assumption of no aggregate features yet). Is this correct?
- However, how does E(N)S(D), i.e. the expected frequency of a ground up loss of size D or greater, relate to obtaining a severity distribution for the layer? Shouldn't it be more related to the frequency distribution?

8. Still in pg 35, one of the process points says "We estimate the aggregate loss for each layer on the exposure rated basis."
- Does this mean obtaining the C_L loss cost for each narrow layer using the original exposure curve (found without any aggregate features yet)?

Section 6.2: Exposure adjustment in treaty experience rating
9. "We can use exposure rates..." - Is the exposure rate just referring to the rates based on exposure curves specifically?

10. The last core reading paragraph: "One way around this is to use historical limits profiles....."
- I am struggling to understand this paragraph, could you please rephrase it somehow?

Sorry for the long questions and thank you!

 

Hi JL24,

Thanks for your questions. I will work through them below:

1. I am sure there is a mathematical derivation, however this is out of the scope of the exam, and as such, I would recommend not worrying about it.
2. Whilst in an ideal world, we wouldn't ignore the open claims, the method detailed on page 29 is the methodology used by ISO. Unfortunately, in the world of insurance pricing, we aren't in an ideal world!
3. This part of the course is talking about the collective risk model - as opposed to individual risks. Depth into why the high RI threshold isn't required here, however in the course notes there are details of a paper by Mack, T and Fackler, M - I recommend you having a look through this.
4. You can think of it like that, yes, but usually Riebesell curves would only be used where ILFs aren't available or can't be derived.
5. This section is essentially explaining the idea that if exposure is used to calculate a reinsurance premium (using exposure curves here is a tool we can use to help come up with a price), then a reinsurance premium can be calculated. But, if there is a reinsurance treaty in place (lots of risks grouped together in some sort of RI programme), then sometimes you can come across some features in the RI programme, called 'aggregate features'. The course notes then explains what some of them are, and how they work. The course notes explain that pricing these aggregate features is beyond the scope of this section, therefore for the SP8 exam, knowing this in detail is not required. Therefore, points 6, 7 and 8 aren't covered here. However, if you would like to know more, page 36 has details of the method in detail: Mata, AJ, Fannin, B and Verheyen, MA, Pricing Excess of Loss Treaty with Loss Sensitive Features: An Exposure Rating Approach, GIRO Convention Paris 2002
6. As above
7. As above
8. As above
9. For this question, and question 10, I recommend reading through the paper, as detailed at the bottom of this section (Mata, AJ and Verheyen, MA), to help solidify knowledge here.
10. As above

I hope this is ok. If not, please do feel free to respond.

Thanks
Aman
ActEd Tutor

 

Hi

For Qn1: G(1) = 1 because the numerator is the LEV at x = 1 is the LEV with no limit which is the same as the expected value (the denominator).

 

Thanks a lot Aman and Busy_Bee4422!

Just a couple of follow-up questions:
Aman - For question 3 above on Riebesell curves, do you mean that Riebesell curves cannot be used for individual risks?

Busy_Bee4422 - If M is just the PML rather than the absolute limit of the risk, Y can be greater than 1 and hence I think LEV at x = 1 would not be equal to the LEV with no limit then? Although intuitively G(1) = 1 still makes sense even if Y can be greater than 1.

 

Hi JL24

We have X such that it is between 0 and M. If you choose M as the PML it seems to me what you have in the denominator of G(x) is the LEV at x =1 rather than the unlimited expected value otherwise you will have an issue of Y being mapped to a larger range of X than the orginal X from which it is derived.

 

Regarding scale invariance, it arises because the curves assume that each time the sum assured doubles the loss cost increases by a fixed percent. This assumption is scale invariant.

 

Thanks a lot Busy_Bee4422!

 

If X is a random variable representing the loss to a property, and M is the measure of the maximum value of X (e.g. Sum Insured), then relative loss severity Y = X/M.

G(x) = {LEV_Y(x)} / E(Y) (eqn 2.1, definition of exposure curve, where x is the deductible as a proportion of max loss)

Substituting the definition of LEV (Limited Expected Value) into the definition of the exposure curve gives:
G(x) = E[min(Y, x)] / E(Y) ……. Equation A

If f(Y) is the pdf of the underlying relative severity curve, then the numerator of equation A is:
E[min(Y,x)] = int(0,1): {min(Y, x)*f(Y)} dY
= int(0,x): {Y*f(Y)} dY + x(1 – F(x) … Equation B

where F(Y) is the cdf of the severity distribution.

When x =1 (eg. deductible is set to sum insured), F(1) =1 (since Y<=1 by definition of Y) so: x(1-F(x)) = 0.

Therefore equation B gives E[min(Y,1)] = int(0,1): Yf(Y) dY = E(Y)

Substituting this into equation A,
G(1) = E(Y)/E(Y) = 1

 

The ISO approach to deriving ILF curves is intended to address the bias towards smaller claims caused by them considering only closed claims, by using sophisticated analysis of claims grouped by weighted payment lags. The course notes give the outline, and the full details are in https://www.casact.org/sites/default/files/database/studynotes_palmer.pdf

 

The 'multiples of 2 of the sum insured' refers to how the risk cost increases by a constant factor 1+z (where z >0) each time the sum insured doubles. From this property, the Riebessel formula for the ILF is derived. But when you are then applying the Riebesell curves to costing of risks, the formula can be applied to any generic increases in limit (not just multiples of 2). See formula 1 "at any sum insured v > 0" on p5 of https://www.actuaries.org/ASTIN/Colloquia/Berlin/Mack_Fackler.pdf

 

Hi

A few thoughts on your other questions

Section 6.1

My big picture understanding of Section 6.1 goes something like this

1. You use equation 2.3 on many narrow strips (of size small delta) of a layer to get the loss costs. (many C_small-delta)

2. You use equation 6.3 on those loss costs i.e. divide the loss cost by the strip size to get an estimate of the left-hand side of the equation. Use these many points to estimate the severity distribution. I take it the E(N) is the same for all so it can be taken as a constant.

3. Make an assumption of the frequency distribution.

4. From steps 2 and 3 above get an aggregate loss model using methods discussed in chapter 11.

5. Use this aggregate loss model to get the impact of the aggregate feature and adjust your rate.

Section 6.2

My big picture understanding of Section 6.2 goes something like this

You may find yourself in a situation where ‘on levelling’ from the ground up may not be suitable. In that event, you can use the loss cost from exposure curves using historic limits as an exposure measure to adjust your experience rate premium for changes in exposure. Basically, put the loss cost from the exposure curve in the column you normally put exposure to calculate your risk premium per unit exposure

 

Hello Ppan13 and Busy_Bee4422,

Thank you both so much for the detailed explanations! Everything is definitely much clearer now.

Just one more thing that I can't quite understand - in Section 6.1 pg 35, how does eqn (6.3) relate to the severity distribution? Since the sentence below it says 'E[N] S(D) represents the expected frequency of a (ground-up) loss of size D or greater', it is slightly confusing that this equation relates to the severity distribution instead.

 

Hi JL24

If you think about it graphically E[N]S(D) is the multiplication of the frequency E[N] with a point S(D) (the survival probability at point D) on the severity distribution curve. The statement is therefore correct.

 

Hi Busy_Bee4422,

Is it right to say that S(D) in eqn 6.3 are the points forming the severity distribution mentioned in 'We use equation (6.3) to estimate a series of points on the severity distribution that is consistent with the exposure rated estimates'?

 

Hi JL24

Yes that's my understanding that you will get may of these E[N]S(Di)s and use them as points to estimate your severity distribution. the E[N] should be the same for all as per our assumptions in section 2.

 

Thanks Busy_Bee4422!