SA3 April 2005, Question 1

[AlphaCharlie]

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!

 

[Ian Senator]

Don't forget the risk is increasing over the period...

So, for a policy written at the start of the year, the total risk 'units' is 1+2+...+24, of which 1+2+...+12 are exposed in the first year, ie 25%.

The rest is earned in the second year, ie 75%.

At 18 months, you've exposed 1+2+...+18 out of a total of 1+2+...+24, which gives you the 56% quoted.

The 6.25% is 25% of 25%, ie combining the calculation in the first bullet point (earnings pattern) with that in the second (writing pattern).

To increase profit by 4m, we therefore need to increase premium by 64m (4/0.0625). So it looks like there's a rogue zero in the examiners' report.

 

[AlphaCharlie]

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.

 

[NeedToQualify]

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

 

[Ian Senator]

Yes, but this is sort of hinted at amongst the comments further down in the report.

 

[Ian Senator]

Oops!

Some eagle-eyed students today in the tutorial spotted where the examiners get their 640m from (cheers guys, well spotted!).

If the premium increased by 640m, then the EP in the current year would increase by 6.25% of this, ie 40m. If the profit margin stays at 10% of premiums (started as 20m out of 200m), then this would give the desired increase in profit of 4m.

So it looks like the examiners' report is correct after all, and I was wrong - sorry!

For those of you with tutorial handouts, note this correction to the solutions pages.

Cheers
Ian

 

[lost_in_sa3]

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

 

[Ian Senator]

Yes, you're being obsessive! But you're correct.

On all these 'earnings patterns' calculations, you can afford to be a little approximate (and not use integration) and make your life a little easier. In reality, of course, there are so many other things you would have to consider which would make the whole thing a little arbitrary anyway.

So make your life a little simpler. It'll also cut down your time in the exam!

 

[Helen.France]

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?

 

[st6student]

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?

 

[Ian Senator]

I think ST6student has a more accurate answer.

Attached back-of-envelope spreadsheet to show his calcs more explicitly, which might help. I've ignored the odd thing that might happen in the very first and last month of each policy (ie assumed everything starts on first of every month).

I think the thing is, whilst we can argue about this until the cows come home, the key thing is that we all realise that the exposure in the first year would only be a very small fraction of the total written premium. Saying this alone without spending ages doing calcs would score sufficient marks to convince the examiner we know what we're talking about, and saves a lot of time. The vast majority of the marks for this question would have been for the rest of the answer discussing whether a large increase in written premium is feasible.

Haven't double-checked the spreadsheet, so if anybody wants to check it, feel free!

 

[st6student]

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!)

 

[Helen.France]

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.