Buhlmann-Straub Assignment X5 Question 1

[jack123]

Question 1 of assignment X5 asks you to calculate a Buhlmann Straub estimate. However the assumptions in the question differ from the assumptions in Chapter 18 Section 5 of the CMP, so I'm struggling to see how the same formula can be applied.

Based on the CMP, for Buhlmann-Straub, we need to know X_ik | theta_i where i is a risk and k is a year. However in the question what we are given is X_i | theta_i.

In addition, the CMP states that var(X_ik | theta_i) needs to equal sigma^2(theta_i)/V_ik. However in the question we are only told var(X_i| theta_i) = 300/V_i


Even if you change the interpretation of i in the question to be a year, the assumptions of the question don't seem to align.

Are the k subscripts missing intentionally or is this a typo?

 

[Katherine Young]

Hi again abc1,

This question only considers the one risk, so it doesn't make sense to have two suffixes. That's why you don't see a k in either the question or solution.

Think of i as relating to the individual risk being considered, as opposed to collateral risks. (As it happens though, we don't happen to make use of any data on collateral risks in this question, so we needn't have used the suffix i at all). I agree this can be confusing after having grown used to the construction given in the Core Reading, but we just have to make the best use of the information we've been given.

Secondly, you ask about the calculation of var(X_ik | theta_i). We need this in order to calculate sigma^2(theta_i)=V_i*[var(X_i | theta_i)]
But since we know that var(X_i | theta_i) = 300/V_i, we can simply substitute this in to the formula for sigma^2(theta_i), to get:

sigma^2(theta_i) = V_i * 300/V_i = 300.

In fact, Buhlmann-Straub questions can appear radically different one from another, but they all use the same fundamental process:
Calculate mu(theta_i)=E[Xi given theta_i]
Calculate sigma^2(theta_i)=V_i[var(X_i given theta_i)]
Use these to calculate:
beta = E[mu(theta_i)]
phi = E[sigma^2(theta_i)]
lambda = var(mu(theta_i)
Use these to calculate the credibility factor, and finish off with your credibility premium.

The trick is to resist being distracted by the questions being presented oddly.

For more practice at questions that don't resemble the chapter, see:
September 2017 Q2 (which is very similar to your assignment question X5.1)
April 2019 Q6 (again similar, but with no suffixes at all this time)
April 2022 Q2 (which involves finding unconditional moments before being able to move on)
Sept 2024 Q7 (again involving calculation of unconditional moments / probabilities).

 

[jack123]

Thanks. Following up on this, I tried September 2017 Q2. Why is it that we use the total volume (320) rather than only the volume for year i (250) when calculating the credibility factor for year i?

In the CMP, we assume there is a claims ratio that depends on a latent parameter theta_i, and then calculate the credibility factor z_i using V_i, the volume of claims for i only.
In the 2017 Q2 question we also assume there is a claims ratio that depends on a latent parameter theta_i. However, here the credibility factor is calculated using total volume V not V_i.

Why the difference?

 

[Katherine Young]

Thanks. Following up on this, I tried September 2017 Q2. Why is it that we use the total volume (320) rather than only the volume for year i (250) when calculating the credibility factor for year i?

The credibility factor is the weight placed on the historical data. So both Z and Xbar needs to be based on total past volume (ie 320), for consistency. 250 is the volume for next year, which has no bearing on the reliability of our past data.

In the CMP, we assume there is a claims ratio that depends on a latent parameter theta_i, and then calculate the credibility factor z_i using V_i, the volume of claims for i only.
In the 2017 Q2 question we also assume there is a claims ratio that depends on a latent parameter theta_i. However, here the credibility factor is calculated using total volume V not V_i.

Why the difference?

The CMP considers more than one risk, and Vi is the volume of claims for risk i. The 2017 exam only considers one risk, so there is no need for a suffix i.


We explain the Buhlmann-Straub method in detail throughout our ASET. Have you considered buying that? Alternatively, you can buy our new product, The Vault, which gives you online access to 10 years of ASETs, with both IFoA and ActEd solutions. You can search by topic to choose which questions you want to practise.

I think it's super! Given the difficulties you're experiencing with SP8, I think you'd find it very useful, I do recommend you buy it.

 

[jack123]

So it seems the formula in the core reading can't always be applied to exam questions.

Core Reading
The assumption in the core reading is that:
E[X_ik|theta_i] = mu(theta_i) where i is a risk

And it states that when the assumption above holds:
z_i = V_i/(V_i + phi/lambda)

This Question
In this particular question, the assumption is:
E[X_ik|theta_i] = mu(theta_i), where i is a year and k is the single individual. So this is the same assumption but with i redefined as a year and k redefined as the risk.

Since the assumption is the same I would have expected the formula for z_i to be the same:
z_i = V_i/(V_i + phi/lambda. I.e. use V_i, the volume for this year only.

But instead the answer is that z_i = V/(V + phi/lambda). I.e. use V, the total volume across all years.

Hope you understand my confusion! We have the same assumption but different formula for z_i.

My guess is that it is because this question is a different scenario to the core reading. In both places we assume that the claims ratio distribution depends on i. However in the core reading our claims ratio estimate is calculated by averaging across k, whereas here we average across i.

I feel like with these questions I'm "fudging" a formula together as we don't have one. But intuitively it makes sense that the credibility factor depends on the volume of data you used to calculate the claims ratio estimate. I wonder if the formula in the core reading is a specific example of a more generalised approach that we are expected to know.


And yes I have the Vault. ASET is usually helpful but in this case it didn't explain why we use the total volume.

 

[Ian Senator]

So it seems the formula in the core reading can't always be applied to exam questions.

Exactly, hence why it's called 'Core' Reading, it gives the basics but can't possibly cover every single scenario or exam question. Indeed, throughout these exams, a key skill you need to develop is the ability to apply/extend the principles in the Core Reading and Course Notes to a new or different scenario. Just like real life!

 

[Katherine Young]

I like your intuitive justification of the method too Jack, it's spot on. For more questions on this, and to see how differently Buhlmann-Straub questions can be presented, see my post earlier in this thread.