September 06 question 11

[Hamilton]

paper http://www.actuaries.org.uk/__data/assets/pdf_file/0012/32304/ct1x_s06.pdf

report http://www.actuaries.org.uk/__data/assets/pdf_file/0017/32336/ct1r_s06.pdf

Hi all , I'm having a spot of bother with this one , and well as always I think I'm right and the exam paper answer is wrong .

So this one is bout a 7 seven year index linked bond that is purchased by an investor after a year and held till maturity i.e. for 6 years and question is at what price for say a £100 nominal . Now this investor is looking for a real net yield of 3% , and inflation is indexed . Ok probably for the best if you read the question from the paper .

Onto my problem , we do a fairly strange thing with inflation in this question , we inflate the coupons and capital and then we deflate ( discount at inflation rate ) them back to present value , strange very !

This isn't a bad thing but I wonder if it has been done at the correct rate

my problem is the use of this term in the answer

0.8 . ( 113.8/110 ) . v / (1+r)^o.5

I would have just used

0.8 . ( 113.2 / 110 ) . v so use the earlier value from our index

the answer in the paper seems to use the 2.5% inflation rate too early ! if that makes sense to anyone , it could be a result of the fact our inflation index is lagged so we might as well take advantage of this functional form inflation rate as early as possible since it is more correct if you will.

But to me this seems like a waste of time since the bond is still lagged by 8 months so its not obeying this 2.5% inflation rate till after the first coupon payment so why include it.

ok to highlight what im taking bout a bit more i would have said the real present value of the second payment is

0.8 . ( 113.2/110 ) . v^2

whereas the report says

0.8 . (113.8/110 ) . v^2 / (1+r)^0.5

i.e. everywhere in answer I would replace (113.8/110)/(1+r)^0.5

with (113.2/110)

so in the report it is inflated one way and deflated another.
Hope I got this all down correctly and in a way people can see what im getting at.

So let me know what you think of this problem . You probably couldn't care less I would imagine , with 4 weeks left to go from today argh

 

[didster]

At a glance, could it be:

1.5%per half year is a REAL rate.
2.5%per annum effective inflation.

Nominal rate (per half year) = (1.015)(1.025)^0.5 ....(1)
Note v in the question is at 1.5%per half year so 1 divided by (1), ie reciprocal, is the same as

v/(1.025)^0.5 ....(2)

First payment is 0.8 (133.8/110) per £100 nominal or £97.31 in (future) £ terms.

Discount this at nominal rate of (1.015)(1.025)^0.5 since it is a nominal £ term at the future date, ie multiply by v/(1.025)^0.5

And so on for other payments.

In theory it should be possible to work entirely in real terms, ie real coupons and redemption (in todays £terms) with real discount rates (although special treatment of first known payment may be needed). In practice, I haven't tried it but you could try it yourself to see if you can do it that way.

 

[Hamilton]

I dont think you get my problem didster

I believe this is wrong

"Discount this at nominal rate of (1.015)(1.025)^0.5 since it is a nominal £ term at the future date, ie multiply by v/(1.025)^0.5 "


since we have inflated over the 6 month period till the first coupon using 113.8/113.2 and then we deflate using 1/(1+r)^.5 which are substantially different , since not much inflation happens in our index in that time .

My point is why do this at all the whole inflation /deflation thing should cancel out and be pointless , no?

 

[Mark Mitchell]

Some initial observations:

Onto my problem , we do a fairly strange thing with inflation in this question , we inflate the coupons and capital and then we deflate ( discount at inflation rate ) them back to present value , strange very

- the examiners' report solution is not "strange". It follows the standard approach to calculating the price of an IL bond based on a real yield.

My point is why do this at all the whole inflation /deflation thing should cancel out and be pointless , no?

- the reason we allow for increasing the payment values for inflation, and then remove inflation again, is because the inflation we put in and the inflation we take out are different. They are different because of the time lag and therefore do not cancel out.

The standard approach to this type of question is:

1. Calculate the actual amounts of payments made under the bond. (Imagine you held the bond and think about the payments that would be credited to your bank account every 6 months.)

2. Write down an equation of value for the bond in "real terms" or "time 0 monetary amounts" ie. by writing down present real values. It makes sense to work in present real values as we are given a real interest rate.

If you get the hang of it for one term in the solution, then the others follow suit. So let's think about the first term.

Step 1.

The first coupon received by the investor is received on 1 Jan 2004. Since coupons are 2% payable half-yearly, the amount of this coupon amount will be 1, with the appropriate amount of inflation added and tax deducted. We need to add inflation over the period since the start of the bond (that's 1.5 years from 1/7/02 to 1/1/04), but also account for the time lag of 8 months.

8 months before the start of the bond is Nov 01 (Index value = 110.0)
8 months before this coupon is May 03 (Index value = 113.8)

So the actual amount received is 1*(113.8/110.0)*0.8 (since tax is 20% on coupons).

Note we're allowing for 1.5 years of inflation, but the period considered is shifted back by 8 months.

Step 2.

The bond pays 0.8*(113.8/110.0) on 1 Jan 04. The present real value of this on 1 July 03 (the date we're interested in) should take account of interest and inflation over the period between these two dates WITH NO TIME LAG.

We need to discount the payment for 6 months. The solution works in terms of a half-yearly effective interest rate, so we need a term of v to account for the operation of interest for six months.

We need a term of (1.025)^(-0.5) to account for the operation of inflation over that period - since we're interested in the period 1/7/03 to 1/1/04, the assumption of 2.5% inflation pa applies the whole time, and we do not need to reference the index.

So the present real value of the first coupon payment is:

0.8*(113.8/110.0)*v*(1.025)^(-0.5)

as stated in the solutions.

Hopefully this helps to explain the method.

 

[Hamilton]

Thanks Mark!

Yeah , thats what I eventual told myself was going on and that I better get used to it.