S.o.s

[AKT]

Hi,

Can someone State and Prove to me (from first principles) the formula for the variance of the Present value of an n-year temporary annuity deferred for m-years?

Thanks in advance

AKT

 

[Louisa]

Hi AKT -
Think this will work with the same method as the answer to Q2.8 in the notes. It doesn't look very pretty tho.
The expression for NPV to start with would be something like
a"_{min(K+1,m+n)¬} - a"_{min(K+1,m)¬}
if that notation makes sense.
Takes about 2 pages, is messy, and I end up with something like (deep breath...)
(1/d^2)* [ (^2A_{x:n+m¬} - A^2_{x:n+m¬}) + (^2A_{x:m¬} - A^2_{x:m¬})
- 2( ^2A_{x:m¬} + v^m _m|A_{x:n¬} + v^{2m+n} _{m+n}p_x
- 2A_{x:n+m¬}A_{x:m¬})]
As you can see, quite a lot of room for mistakes there, so that's not guaranteed!
Note also the helpful note at the bottom of Solution 2.8 - thankfully we don't have to be able to reproduce the whole of one of these.
Louisa

 

[AKT]

Thanks very much Louisa,

Actually i started exactly the way you did, trying to get the Expected values of the annuities but got some covariance on the way that I couldn't isolate or bring to a better format, It became so messy I had to leave it...
I hope, no one will think of asking such question at the exam..

Cheers,