Immunisation April 2005 Q8

Part II is what is causing me trouble, I am following the standard différentiation of the PVs of Assets and Liabilities but it seems in the answers they forget to bring the power of v down by 1? I diff with respect to i but I still got the 2nd condition as not being satisfied...

Official Answers:



Update Ok so it appears their differentiation w.r.t.delta is correct so I must have gone wrong somewhere!

 

As far as I'm aware the power of v should go up when you differentiate w.r.t. i. Because v = (1+i)^(-1), and when you differentiate that you get -(1+i)^(-2) = -v^2, and then v^3 and so on.

HOWEVER, if you are equating the PVL' and PVA', then it doesn't matter if they are all off by a factor of v. Because the value of i is the same for both sides, and so you can divide by v on both sides, or whatever you want to do and it won't affect the equation.

In other words

Av^14 + Bv^16 = Cv^27 + Dv^12

can also be written as

Av^13 + Bv^15 = Cv^26 + Dv^11

Therefore I think your calculation may be off somehow, because your query shouldn't affect the equation.

Look in the notes below where it lists Redington's conditions, below it is a series of other conditions that are equivalent to condition 2 being satisfied.

Maybe if you show your working we can have a look at what you're doing and maybe see where you're going wrong.

 

Thanks

The method in the solution set is perfectly correct since there is a direct relationship between i and the force of interest, we can differentiate the PV of assets and liabilities with respect to delta. Since (1+i)=e^delta we have v^n=(1+i)^-n=e^-n*delta and as we known differentiating that just gives-ne^-n=nv^n which is what they've done. It's pretty neat in a way I guess as for a lot of questions that are slightly more algebra dense it makes things considerably easier.

 

Hmmm

Good point, I figure their method is still right even if they don't include it in the core reading. What method would you favour then? For speed and accuracy instead? You'll need convexity so I guess you'll need the first and second derivatives of the PV values anyways.

 

I can't quite see how writing it as a sum makes it easier?

E.g. if we have an annuity that pays 5 in arrears for 10 years, what's the best way for finding the DMT?

If I write it out as a sum, I get 5v + 5v^2 + 5v^3 + ... + 5v^10, and differentiating I get 5 + 10v + 15v^2 + ... + 50v^9. But surely this can't be equated into a simple geometric progression? I would say it's easier to write out the annuity as (1 - v^10) . i^-1 and then use the quotient rule for differentiation. But maybe I can't see the glaringly obvious..


edit: in fact, I guess I could write it as a sum of a level annuity and an increasing annuity! And hence I see why you wrote that in your post! Uff..at least I realised for myself

edit again: oops, I see that I differentiated wrt v and not i. But I guess the principle is still there.

final edit (I promise!): Am I right to assume all level annuities of coupon D (ignoring redeemable value) of time T, differentiate to Dv (Ia)T| ?