september 2006 q11 - variance of annuities

Part iii asks for the variance of the annuities.

the solution evaluates the variance for each annuity as E(X^2) - E(X)^2. To do so it calculates the first term using an integral and gets the second term from part ii.

i attempted to calculate the variance by writing the PV of the annuity as 5000 * ( a_Tx - a_min(Tx, 2) ). i could then evaluate the variances of these using the formulae given in the Tables. I get a different to the one in the solutions, so i was curious whether it was because my calculations were incorrect (i cant see that they are) or the methodology was incorrect.

my question really is should i be able to work out the variance as ive described, or is that incorrect?

thanks

 

I think it's got to do with the fact that you had min(Tx, 2) instead of just a_2. This will reduce your variance to var(a_Tx) though since a_2 has no variance, which doesn't match with the case study.

Unlike the discrete version for variance of a deffered annuity (see Question 2.8) I found this question hard(er).

 

I think this expression in terms of RVs looks fine.

I suspect your problem comes when taking the variance of this. It's effectively of the form var(X+Y), but since X and Y are not independent RVs in this case (they both depend on the distribution of Tx), then you'd need to include a covariance term. And I think that would be nasty to evaluate.

 

ah ok that makes sense. thanks mark.

and edwin yes i find the variance ones a bit tricky to deduce the correct formula to use straight away. furthermore i find some past papers are a bit more leniant than others when it comes to the definition of the word "derive" (not in this example though). hopefully if any come up theyre straight forward!

 

Mark this solution then lacks only the co-variance term, that should mean it was possible to get some credit for the Var(X) + Var(Y), am I right?

 

Yes - part credit is given for partly correct answers.