CT5- April 2017, qus10

[Bharti Singla]

Hii all
I am struggling with the deferred annuity part of this qus. It is not clearly mentioned whether the annuity of £10,000 pa is payable until joint lifetime or until last survivor?
Also, not getting what they have done in the solution.
Could anyone please help?

Thank you

 

[Robert Chadburn]

I think the wording is clear. It says 10000 pa while both are alive (which is the same as saying "while the joint life status is active") and 5000pa while only one is alive (this means we are paying 5000pa for the lifetime of the survivor following the first death). This is clear and unambiguous to me - I'm not sure how else you could interpret it?

Getting their formula directly is quite a leap though. However, we should be able to confirm their answer through general reasoning. They have given the value as:

5000x( {deferred annuity payable for lifetime of of male} + {deferred annuity payable for lifetime of female})

So if both lives die before time 10, nothing is payable = correct.
If just one of the lives survives beyond time 10, then 5000pa starts from time 10 and ceases on that person's death = correct.
If both survive 10 years, then 10000pa starts from time 10. At the point at which either life subsequently dies, then one of the annuities ceases and the other carries on - ie following the first death 5000pa continues until the second death = correct.

Adding up all the above possibilities gives the required payment pattern.

In the exam you may not spot this. However, you can get the same answer by explicitly valuing each possible outcome.
(1) P(Male survives 10 years) x P(female dies in 10 years) x EPV of annuity for 5000 starting at time 10 to male.
(2) same as (1) but lives swapped.
(3) P(Both survive 10 years) x [{EPV of joint life annuity for 10000 starting at time 10} + EPV of Revy annuity for 5000 starting at time 10 to female after death of male + EPV of Revy annuity for 5000 starting at time 10 to male after death of female. }]

Note for others reading this - this is explained in ASET as well.
Good luck Bharti (and everyone)

 

[Kanishka]

I too have a doubt in this
I initially solved this using the parts (1) (2) and part 1 of (3) and got the wrong answer. But why must we include the 2nd n 3rd parts of 3.Isn't it already covered by (1) and (2)?

 

[Robert Chadburn]

No, because in the second part of (3) the reversionary payments are only those that would occur given that BOTH are alive at time 10.
In (1) and (2) only one of the lives is alive at time 10.

Basically we have four conditions to consider:
(1) payments conditional on just the male being alive at time 10
(2) payments conditional on just the female being alive at time 10
(3) payments conditional on both being alive at time 10.
(4) payments conditional on neither being alive at time 10
Of course the last one has zero value, and so we ignore it. But all four events constitute mutually exclusive and exhaustive events (their probabilities add to 1). There is no overlap between them (ie they are mutually exclusive) and so we need to include them all, there can be no double counting.

Robert