CT5 Chapter 2

Hi,

I have a query in Q 2.8, in chapter 2, on page 22.

Can anyone explain me me the derivation that follows the expression?

Thanks for your help!

Regards,
Shyam

 

Expression \( V^{n} \require{enclose}\ddot{a}_{\enclose{actuarial}{max\left(K_{x}+1-n,0\right)}}\)
is annuity deferred for n years. The present value is maximum of for duration Kx+1-n or 0.
If life survives for n years then \(K_{x} + 1 - n\) and if dies before n then 0.
Here you can think whole life annuity minus temporary annuity for n years.
Now coming to derivation.
4th line - the term inside bracket \(V^{max \left(K_{x} + 1 - n + n, 0 + n\right)} / V^{n} \)
the rest follows simple math.
Now present value random variable given on next page.
it is assurance same like guaranteed annuity for n years. So if life dies before n years or after n years there is benefit.