SA3 April 2005, Question 1

Hi guys.

I'm having a real problem trying to understand one of the SA3 exam paper solutions, and I was hoping someone could shed some light.

It relates to Question 1(i) of SA3 April 2005.

The policies have 2-year exposures and are written uniformly throughout the year.

The examiners' report states:


"For two year policies: earned in yr 1 = 50% * 1yr * 50% pol exposure =
25%. Therefore 75% earned in year 2. (At 18 mths = 50% * 1.5yr * 75%
pol exposure = 56% )"

I'm not sure I understand this calculation.

It then goes on to state:

"Policies to be written in the forthcoming year will commence uniformly
over the year and will on average be written half way through the year.
They will thus be exposed for a quarter of their policy term. In the
following year the earned premium on these policies will thus be 6.25% of
that written."

Again, I don't understand this. I understand that the policies are exposed to a quarter of their terms, since the average policy is written halfway through the financial year, and hence there is 0.5 years' average exposure, out of a 2-year exposure period. But how do we go from this to 6.25%?

The solution continues:

"The portfolio premium would need to grow by $640m."

Should this say $64m?

Can anyone help??

Thanks!

 

Argh! Thanks for this, Ian.

Classic "read the question" error: I read it as being a uniform exposure over the period, rather than exposure increasing from zero at a uniform rate.

Excellent explanation...very much appreciated.

 

Also..doesn't the solution ignore any additional claims coming from these additional premiums? i.e. it assumes that the LR will decrease!

 

still not convinced about the 6.25% bit

Sorry for being a bit obsessive about this problem -- I understand that in the specific case an approximate solution might be all you need -- but I'm still not convinced about the 6.25% (=1/16) bit.

I think that the earned premium in the first year should be 1/12 rather than 1/16:

(a) A policy starting at time t will have a cumulative risk of (1-t)^2/4 in year 1.
(b) The amount of premium written in interval (t, t+dt) will be P*dt (=P once integrated between 0 and 1).
(c) Therefore the earned premium between 0 and 1 is the integral between 0 and 1 of P*(1-t)^2/4*dt = P/12.

The solution in the report seems to rely on the idea that since policies are written uniformly over the year you will get the correct result by assuming that all policies are written in mid-year -- and a mid-year policy indeed contributes for (1/2)^2/4 = 1/16 of the premium – but this doesn’t seem correct to me when risk is not uniform. Think of this "extreme" situation: what if the risk density were 0 for the first six months, and uniform after that? The above approximation would conclude that the premium earned in year 1 is zero!

However, I might be missing something, or maybe I'm answering the wrong question. Could someone please comment on this?

Cheers

 

Right I'm nearly there but what I don't understand is why when we assume that the average policy is written half way though the year we consider the total risk earned in the whole first year?

My first thought would be to assume that if the average policy is exposed for the first six months it therefore is exposed to 1+2+3+4+5+6 risk 'units'.

but this in itself doesn't seem appropriate because the pattern is not uniform so the half the policies written in the first half of the year would earn proportionately more risk that those in the second half of the year?

So not sure what I would propose as the right idea but surely it's clear that only the few polcies written at the start of the year earn the full first twelve risk units and so to assume the sum from 1 to 12 seems to overestime exposure?

Woud you lose marks for doing the risk profile in half year profiles ie 1 unit in Yr 1 HY1, 2 in Yr1 HY2, 3 in Yr2 HY1 and 4 in Yr2 HY2 so that 30% of the risk is earned in the first year? Clearly this is more of an approximation?

 

I agree with Helen - doesn't it seem a bit odd to use the 1+2+.. thing for one policy, but then ignore the increasing risk for when it actually gets earned?

Couldn't you say something like, for policies written in a year, look at month they're written in:
Month Earned
1 78
2 66
3 55
4 45
5 36
6 28
7 21
8 15
9 10
10 6
11 3
12 1

For risk units earned in each month (assuming they're written at the start of each month, say) - to give total risk units earned of 30.33 (total of above / 12) - out of total risk units of 300 - so 10.1% earned in first year?

Also, when they say 640 increase because 6.25% earned in year written and 6.25% * 640 gives the required 40 increase in total earned premium: Surely total earned premium includes some of the premium earned from policies written in the previous 2 years which is less than the full 200 - so a substantially bigger increase is needed.

Would have thought total earned premium = 6.25% * new rate + (1 - 6.25%) * 200 [the (1-6.25%) representing the unearned part from time 0.5-1.5 for policies written previous year and from 1.5-2 for the year before that]. Putting this equal to 240 (required GEP) gives new rate needed = 840?

 

Thank Ian - was going to ask whether these kind of calcs are really needed as it looks from the examiners' report like only a couple were for the actual figures - and it's pretty obvious that a pretty huge rate increase is needed without calculating the figure. But you've answered that for me now

Having said that, I don't think it's that obvious whether the amount of this big increase is going to be a doubling, tripling, whatever of rates, and so quite how (un)achievable it would be. Seems like their figure is quite materially different from even a decent 5 min "back of the envelope" stab at it. Seems if you were trying to advise in "real life" and advised you'd need to roughly triple rates, looked at it again and realised you'd actually need to quadruple them, the people you're advising may not be too happy (not exactly a minor/rounding error!)

 

On your second point if you assume that a certaint amount of new business would be generated in line with prior years then you could say that its 640 new policies over and above the normal level of new business.