ST7 Prem Rate Change from Underwriting year to Accident Year

[ngongtiti]

Hey, I was wondering can someone help explain how we convert premium rate change from an underwriting basis to an Accident year basis?
Example: let say I have a Loss ratio (Accident year basis) and rate change (Underwriting year basis) for the following periods, to be used to project the 2004 loss ratios
Jan 2003 =5%
July 2003 =6%
Jan 2004 =-8%
Loss Ratio for 2003 =73%
How do I estimate my cumulative rate change factor on an accident year basis that will be used to adjust the 2003 loss ratios??

 

[Katherine Young]

That's rather a convoluted question ngongtiti! In fact, it's not so easy as you suggest, because rate changes apply to written business not earned business. So we have to back out what loss ratio the business was written at, and then work out which segments of business are earned in the period, and at what rates.

We're told 2003 AY loss ratio=73%.

This loss ratio could arise from business written in 2002, or business written in 2003.

Let’s say the loss ratio for business written in 2002 is x% and WP in 2002=100. So 50 of this will be earned in 2003.

Now say we write 50WP in the first 6 months of 2003. For business written between Jan and end of June, the LR is x/1.05, and on average this business will be 75% earned by the end of the year.

Say we also write 50WP in the last 6 months of 2003, the loss ratio is x/(1.05*1.06), and this will be 25% earned by the end of the year.

So our total EP for 2003 is 50+0.75*50+0.25*50=100, and our total claims for AY 2003 are 50*x+50*75%*x/1.05+50*25%*x/(1.05*1.06)=50x(1+0.75/1.05+0.25/(1.05*1.06))

So our 2003 AY loss ratio = claims/EP=50x(1+0.75/1.05+0.25/(1.05*1.06))/100=73%

Solving for x gives our loss ratio for underwriting year 2002: of x=75.3003% …

… And the loss ratio for business written in the first 6 months of 2003 = x/1.05=71.715% …

… And the loss ratio for business written in the last 6 months of 2003 = x/(1.05*1.06)=67.655% …

… And the loss ratio for underwriting year 2004 = x/(1.05*1.06*0.92)=73.538%.

So, to answer your original question, let’s say you need to estimate the loss ratio for accident year 2004:

Claims occurring in 2004 can arise from business written in the first 6 months of 2003 (50WP, of which 12.5 will be earned in 2004), and from business written in the last 6 months of 2003 (50WP, assuming 37.5 to earn in 2004), and from business written in 2004 (100WP in 2004, of which 50 to earn in 2004).

So total EP in 2004 is 100, and 2004 claims are 12.5*71.715%+37.5*67.655%+50*73.538%=71.104.

So the 2004 AY loss ratio = 71.104%.

… all rather spurious of course, it depends on how much business is written in each 6 month period, not to mention claims inflation, earnings patterns, changes in terms and conditions…

 

[ngongtiti]

Many thanks for that. That's certainly abit more complicated than I had hoped for. This is part of the ST7 notes as below:


Premiums


Where premiums are used in the loss ratio calculation, the type of premium to

use will depend on the claims cohort. Earned premiums are appropriate for an

accident year cohort. Written premiums are appropriate for an underwriting year

cohort. A reporting year cohort would be very difficult to use in a loss ratio

calculation, because premiums and claims would have to be found with the

corresponding exposure.

The premium that is consistent with the claims data should be used; for

example, if claims information is provided on an accident year basis, earned

premiums should be provided or written premiums should be converted to

earned premiums.

 

[Katherine Young]

Complicated, certainly. But exam standard? Pretty much.

In fact, your question is from an old 403 exam, so it's certainly exam standard for SA3, and similar sorts of questions (perhaps not quite so tricky) also come up in ST7.