2021 April past exam Q6(ii)

[yuli2513]

Hi,

I do not fully understand why the term assurance component and the deferred whole life assurance component are independent of each other. Either the 40000 under the term assurance component or the deferred whole life assurance component will be paid out eventually. So I would consider them dependent as one must be zero and the other non-zero.

In my understanding, this is the same as var(H) does not equal to var(F) + var(G) in chapter 15 where F (term assurance) and G (pure endowment) are not independent. Here H is the corresponding endowment assurance.

Could you please kindly help me to figure out where my thoughts went wrong? Thanks a lot!

 

[Joe Hook]

Hi,

The two contracts are not independent as you say so we haven't applied the relationship Var(term assurance) + Var(deferred whole life assurance). Instead we've worked from first principles to calculate the overall EPV(benefits at squared rate of interest) - EPV(benefits).

We can demonstrate this with numbers.

Var(term assurance) = E[term assurance @ squared interest] - E[term assurance]
= 29,538,299 - 1206.91^2
= 28,081,665

Var(deferred whole life) = E[deferred whole life @ squared interest] - E[term assurance]
= 58,146,408 - 6462.86^2
= 16,377,829

The two added together equals 44,459,494 which is not equal to the variance from the solution. You'll recognise some of the numbers from the solution as we do sum the second moments of each "product" and so the difference is in the combination of the benefits together for the purposes of calculation the expectation squared.

 

[zubbo]

Hi,

I'm having issues with the question as well. I tried to another way i.e. split benefit as whole whole life assurances:
1- 40K A45
2- 10K A60 which is adjust for TVM (v15) and survival (15p45)

The EPV comes out the same (0.30 diff which I assume is attributable to rounding) but the variance is not the same, not sure what I'm missing here.

 

[Joe Hook]

EPV(benefits squared) will involve a 40000^2 term for the first 15 years and 50000^2 for the 15 years beyond that. Biggest issue with your idea is that 40000^2 + 10000^2 does not equal 50000^2. So unfortunately you're not calculating the second moment correctly in that way.