September 2008, Question 11(i)- Redington Question

[trackr]

Hey guys,

This question is throwing me off a little bit.

So the question states that a company has a liability of $400,000 in 10 years time. The effective interest rate is 8% p.a.

So value of liabilities = 400,000 * (1+8%)^(-10)

Now in the solution they have written:

V'(L) = 400,000*(10)*(1+8%)^(-10)

While solving I had written:


V'(L) = 400,000*(-10)*(1+8%)^(-11)


So where am I going wrong? I don't seem to understand the solution. Any help is much appreciated.

Here is the link for the past exam paper:

http://www.actuaries.org.uk/researc...subject-ct1-exam-papers-and-examiners-reports

It is on page 137 of PDF file.

Thanks.

 

[morrisja]

I suppose your question is about calculating the DMT of the Liabilities then?

Can you explain where you got your version from? (That should show you why it's not right).


We're basically looking for the DMT of a ZCB. Which we know is the PV of the payment * time to maturity of the bond. (Divided by the price, but that's irrelevant here as the first condition means V(A) = V(L) so price doesn't need to be factored into the DMT and V'(A)=V'(L) is enough for DMT equality)

So we have the PV of the Payment:

400,000*(1.08)^-10

And the TTM of the Liability:

10

Yielding:
4,000,000*(1.08)^(-10)

Hopefully this helps. If it's not clear (or I've made a mistake), let me know.

 

[trackr]

Hey Morris,

Thanks for replying.

Well the way I calculated was basically differentiating the present value of liabilities with respect to i to get V'(L). That's how I thought it was. I have solved a few questions like that and seem to be getting the right answer. It is just this question.

Does that answer your question?

I really don't understand why we just raise the interest rate factor by -10 instead of -11.

I know there is something very very silly and obvious. But I am just not seeing it. I have been solving so many exam papers. I think my brain is dead.

 

[morrisja]

Apologies, your idea isn't wrong - and my reply is badly put together.. I'd completely forgotten about differentiating WRT i..

If you apply that to the assets also - does it not result in the same answer?

They have applied the same technique to both sides. If you look at their ZCB it's 12v^12 - I presume you would have gotten -12v^13

I found that differentiating WRT the interest rate is tedious and open to error, so I would always have used this method.

 

[trackr]

Apologies, your idea isn't wrong - and my reply is badly put together.. I'd completely forgotten about differentiating WRT i..

If you apply that to the assets also - does it not result in the same answer?

They have applied the same technique to both sides. If you look at their ZCB it's 12v^12 - I presume you would have gotten -12v^13

I found that differentiating WRT the interest rate is tedious and open to error, so I would always have used this method.

Oh yes, I made a mistake on my solution and when I checked the examiner's report solution I thought my approach was wrong.

Thank you, so much. I knew it was something silly.

Cheers!

 

[morrisja]

Good stuff, I wouldn't be a huge amount of help with the differentiating approach anymore, but there's nothing wrong with the approach as long as you're comfortable with it.

Good luck with the exam.