Yes, this is still relevant. The CAS tests the Fellowship exams at a higher level of Bloom's taxonomy so you're required to try and work out what the exam committee were thinking. In this case, they were asking for the discount in expected premium as a result of choosing a retro policy over a guaranteed cost policy.
If in doubt, state your assumptions clearly and make sure to look at the point value of the question to check you're not oversimplifying something.
We're told this is a balanced plan so the expected retrospective premium should cover both the expected loss and expenses, i.e. E[R] = E+e
The CAS solution lets D be the discount for the retro plan relative to the standard premium, so (1-D)*SP is the expected retro premium, E[R]. Since this is a balanced plan we can jump directly to E+e = (1-D)*SP.
You'll get the same answer if you find E[R] = (B+cE[L])*T (T=1 in this question) and then compute E[R]/SP -1.
I am struggling to understand what I am missing when I calculate the insurance charges at the max and minimum loss. I correctly identified the points. Take the charge at rg: the CAS solution is as follows:
Isn't the top part of the equation just a box? I would have thought it should have been a triangle since the distribution is uniform so the area above the max loss of 750k would be .5*250k*.25 (aka half their answer). I am sure there is a trick I am missing here.
Beyond that, I think I am missing why I need to divide by the expected loss. I assumed I could just calculate the area of the charge directly but I believe I am missing something by attempting to go that route.
When working with entry ratios it's important to remember they are a ratio to E[A] (or E[A_D] if there is a per-occurrence limit). This is why we divide by the expected loss.
The charge at r_G is given by the area of the following triangle. Letting F(x) be on the x-axis and loss $ on the y-axis then the triangle is from 750k to 1m on the y-axis. If you compute the area of the triangle as 0.5 *(1,000,000 - 750,000)*( 1 - 750,000/1,000,000) you get 31,250 which is the dollar amount of the insurance charge, i.e. E[A]*phi(r_G). Dividing by E[A] gives the correct answer of 0.0625.
If you convert the y-axis to entry ratios first by dividing through by E[A] then you get the figure you provided by factoring out the 1/2 from the numerator and moving the 500,000 from the denominator into the denominator of the first term in the numerator.
Upshot is try to keep tabs on whether you're working in dollars or in terms of entry ratios and avoid mixing the two.
Comments
Yes, this is still relevant. The CAS tests the Fellowship exams at a higher level of Bloom's taxonomy so you're required to try and work out what the exam committee were thinking. In this case, they were asking for the discount in expected premium as a result of choosing a retro policy over a guaranteed cost policy.
If in doubt, state your assumptions clearly and make sure to look at the point value of the question to check you're not oversimplifying something.
Ok, so my assumption would have been that we want D where SP/R - 1 = D (R is retro prem).
I don't see this as an acceptable solution if I look at the CAS solutions. Why is e+ E = (1-D)*SP?
Also, would it be possible to solve the equation I provided? I tried, but got stuck.
We're told this is a balanced plan so the expected retrospective premium should cover both the expected loss and expenses, i.e. E[R] = E+e
The CAS solution lets D be the discount for the retro plan relative to the standard premium, so (1-D)*SP is the expected retro premium, E[R]. Since this is a balanced plan we can jump directly to E+e = (1-D)*SP.
You'll get the same answer if you find E[R] = (B+cE[L])*T (T=1 in this question) and then compute E[R]/SP -1.
I missed that somehow. I think I got fed up and gave myself a mental block with this chapter lol.
I am struggling to understand what I am missing when I calculate the insurance charges at the max and minimum loss. I correctly identified the points. Take the charge at rg: the CAS solution is as follows:
Isn't the top part of the equation just a box? I would have thought it should have been a triangle since the distribution is uniform so the area above the max loss of 750k would be .5*250k*.25 (aka half their answer). I am sure there is a trick I am missing here.
Beyond that, I think I am missing why I need to divide by the expected loss. I assumed I could just calculate the area of the charge directly but I believe I am missing something by attempting to go that route.
Thank you in advance!
When working with entry ratios it's important to remember they are a ratio to E[A] (or E[A_D] if there is a per-occurrence limit). This is why we divide by the expected loss.
The charge at r_G is given by the area of the following triangle. Letting F(x) be on the x-axis and loss $ on the y-axis then the triangle is from 750k to 1m on the y-axis. If you compute the area of the triangle as 0.5 *(1,000,000 - 750,000)*( 1 - 750,000/1,000,000) you get 31,250 which is the dollar amount of the insurance charge, i.e. E[A]*phi(r_G). Dividing by E[A] gives the correct answer of 0.0625.
If you convert the y-axis to entry ratios first by dividing through by E[A] then you get the figure you provided by factoring out the 1/2 from the numerator and moving the 500,000 from the denominator into the denominator of the first term in the numerator.
Upshot is try to keep tabs on whether you're working in dollars or in terms of entry ratios and avoid mixing the two.