Oct 2012 Ques 11

[Pratik]

My logic to this has been 500{ a(5 fixed term) + 2/3 * 5|a(xy last survivor) + 1/3 * 5|a(xy joint life)}
This is because:-
1. for the first 5 years its a fixed payment of 500 irrespective of deaths
2. 2/3 incase any of the lives are alive after 5 years. Therefore the annuity function caters to either life being alive and this parameter is alive. On both dying it becomes zero. Therefore irrespective of male or female being alive this parameters ensures that annuity still continues.
3. 1/3 incase both the lives are active
My assumption is that this satisfies the question requirement.

However, in the solution provided there are two more parameters getting added which exclusively looks at either of the lives dying and 2/3 still getting paid out.

My question is that are these two additional parameters not being catered in the last survivor portion???

Kindly assist.

Thanks and regards,

Pratik

 

[Robert Chadburn]

Your expression is not identical to the first expression that is given in the (IFoA) solution. In their solution, BOTH LIVES HAVE TO SURVIVE to time 5 before the last-survivor and joint-life annuities start. So we will need separate expressions to cover the cases where either (1) just the male survives 5 years or (2) just the female survives 5 years.
In your solution you use the deferred notation symbol, and is (also) correct. If you work out 5|axy-bar it splits into:

(v^5)(5Px)(5Qy)[ax+5] + (v^5)(5Py)(5Qx)[ay+5] + (v^5)(5Pxy)[ax+5:y+5-bar]

while 5|axy is just (v^5)(5Pxy)[ax+5:y+5]

(This formula exactly accords with your verbal explanation.)
And so you see you get the same answer as the given solution.
Robert