Option Pricing calculation

[Trevor]

Hi, I am trying to understand the option pricing calculation in Question 2 of ST1 2008 April past paper.

I am assuming the North American part of this question is not examinable (for question iii at least) as it is mentioned very briefly in the notes, under the ActEd text rather than core reading.

However I am trying to understand the calculation for the conventional method.
Specifically, it is the end of page 5 of the examiner report, why is the PV premium exercising the option S*A[x+10]:10*ax+10:10/a[x+10]:10?
(In my formula, A[x+10]:10* is the term assurance factor, no maturity benefit)

As a term assurance product, the premium calculation will be:
Pa[x+10]:10= SA[x+10]:10 for new policies (ie: not extending the contract);
Pa[x+10]:10= SAx+10:10 for those who extends the contract

if we rearrange the formula for those who extend, wouldn't it be P= SAx+10:10 /a[x+10]:10 ?

1. Why is the solution using the select assurance factor for S?
   My understanding is that for the conventional method, the mortality rates of those who exercised option should be ultimate selection, which is heavier due to antiselection. so it should be using Ax+10:10 instead of A[x+10]:10.
   
   
2. And then why is it multiplied by ax+10:10 ? I am struggling to understand where does this term comes from.

Can anyone explain this part to me?

Thanks,
Trevor

 

[Mark Willder]

Hi, I am trying to understand the option pricing calculation in Question 2 of ST1 2008 April past paper.

I am assuming the North American part of this question is not examinable (for question iii at least) as it is mentioned very briefly in the notes, under the ActEd text rather than core reading.

However I am trying to understand the calculation for the conventional method.
Specifically, it is the end of page 5 of the examiner report, why is the PV premium exercising the option S*A[x+10]:10*ax+10:10/a[x+10]:10?
(In my formula, A[x+10]:10* is the term assurance factor, no maturity benefit)

As a term assurance product, the premium calculation will be:
Pa[x+10]:10= SA[x+10]:10 for new policies (ie: not extending the contract);
Pa[x+10]:10= SAx+10:10 for those who extends the contract

if we rearrange the formula for those who extend, wouldn't it be P= SAx+10:10 /a[x+10]:10 ?

1. Why is the solution using the select assurance factor for S?
   My understanding is that for the conventional method, the mortality rates of those who exercised option should be ultimate selection, which is heavier due to antiselection. so it should be using Ax+10:10 instead of A[x+10]:10.
   
   
2. And then why is it multiplied by ax+10:10 ? I am struggling to understand where does this term comes from.

Can anyone explain this part to me?

Thanks,
Trevor

Hi Trevor

All parts of this question are beyond the syllabus as it stands as the Core Reading does not describe two methods (in the past the conventional and North American methods were explicitly described in the course). So this question could only be examined now if the exam question first described the conventional and North American methods - this would be ok as the Core Reading does cover option take up rates (the North American method) and the alternative of everyone accepting the option (the conventional method).

I think you have the notation for term assurances and endowments the wrong way around. The formulae you have mentioned above all relate to endowments and would have a maturity benefit. By putting a little one above the age we are referring to a function that only pays out on death (and in this case critical illness).

You've used ultimate mortality for the premium calculation which is also wrong. The premium when the option is exercised should use select mortality. This is the point of mortality/morbidity options - the policyholder pays the same premium that they would have paid if they took out a new policy with underwriting, even though they haven't been underwritten. This of course leads to anti-selection and the need to charge an extra premium on the original policy to cover the cost of the option.

This should give you enough to confirm that the examiners solution is correct.

Best wishes

Mark

 

[Trevor]

Hi Mark,

I am aware that the assurance function needs a "1" above the age, as mentioned in my post

(In my formula, A[x+10]:10* is the term assurance factor, no maturity benefit)

I just don't know how to type it out in the textbox here.

You've used ultimate mortality for the premium calculation which is also wrong. The premium when the option is exercised should use select mortality. This is the point of mortality/morbidity options - the policyholder pays the same premium that they would have paid if they took out a new policy with underwriting, even though they haven't been underwritten. This of course leads to anti-selection and the need to charge an extra premium on the original policy to cover the cost of the option.

After seeing the solution again, I realise I confused between the "cost" and "premium":
The "cost" should be based on ultimate mortality because this is what we will realistically expect the experience to be.
Whereas the "premium" should be based on select mortality, because this is the intention of the guaranteed renewability option without underwriting.

Therefore the cost of the benefits is supposed to be : Pax+10:10= SAx'+10:10
(where x' represents the little "1" above x)
and the premium basis will be: Pa[x+10]:10= SA[x'+10]:10.
The difference between these 2 makes the (additional) cost of the option.

Relooking at the formula again, I think I now know where does the " *ax+10:10 " comes from:

The parts not highlighted, P= S*A[x'+10]:10/a[x+10]:10 represents the annual premium we are charging.
We want to evaluate the PV cost (instead of regular premium) as at the option exercise date, therefore, the LHS and RHS of the equation multiplies by ax+10:10 (highlighted yellow) as this is the best estimate basis.

And then the next line after this:
S* [ Ax'+10:10 - A[x'+10]:10/a[x+10]:10 ]
is to get the premium difference the option makes

And this difference gets discounted back to the inception date, allowing for survivorship to arrive to the final premium.

Is this the correct explanation?

 

[Mark Willder]

Hi Mark,

I am aware that the assurance function needs a "1" above the age, as mentioned in my post

I just don't know how to type it out in the textbox here.



After seeing the solution again, I realise I confused between the "cost" and "premium":
The "cost" should be based on ultimate mortality because this is what we will realistically expect the experience to be.
Whereas the "premium" should be based on select mortality, because this is the intention of the guaranteed renewability option without underwriting.

Therefore the cost of the benefits is supposed to be : Pax+10:10= SAx'+10:10
(where x' represents the little "1" above x)
and the premium basis will be: Pa[x+10]:10= SA[x'+10]:10.
The difference between these 2 makes the (additional) cost of the option.

Relooking at the formula again, I think I now know where does the " *ax+10:10 " comes from:

The parts not highlighted, P= S*A[x'+10]:10/a[x+10]:10 represents the annual premium we are charging.
We want to evaluate the PV cost (instead of regular premium) as at the option exercise date, therefore, the LHS and RHS of the equation multiplies by ax+10:10 (highlighted yellow) as this is the best estimate basis.

And then the next line after this:
S* [ Ax'+10:10 - A[x'+10]:10/a[x+10]:10 ]
is to get the premium difference the option makes

And this difference gets discounted back to the inception date, allowing for survivorship to arrive to the final premium.

Is this the correct explanation?

Hi Trevor

Yes, this looks correct.

I'm sorry for misinterpreting your comments on the term assurance functions. Check out the Examination Handbook on the IFoA website for the suggested notation to use in the exam.

Best wishes

Mark