CT1 - Question 3.8

[ssaini]

Hi everyone!

I just needed a little help..

I was working on question 3.8 from the combined material pack CT1
which is as follows:

if the force of interest is delta(t)=0.08+0.02/(t+1), (t>=0) calculate,
(i) the accumulated value at time t=5 of an investment of £1000 at time t=0

and to apply a rasonable check to support the answer.

i managed to do the initial part and was looking over the solution "for the reasonale check part" and couldn't get my head around it.

It says the force of interest over the period averages around 8.5% which is calculated for t=2.5 but the time period is 5??

Any help as to why this is will be much appreciated.

Thanks in advance for your help

 

[Calum]

The 2.5 (I presume) is used to calculate the interest rate at t=2.5, which is a reasonable value to take as an "average".

You then repeat the calculation using this fixed force of interest for the whole time period.

 

[ssaini]

Thank you for your reply.

If we know the time period is 5 and we are given delta(the force of interst), why would we need to calculate an average of half the time period t=2.5. And why would it be reasonable to take 2.5 as an average and say why not 3 or 4?

 

[Calum]

The force of interest is changing monotonically at all times, starting at 0.09 at t=0 and drifting down to 8.033.. at t=5. Using t=2.5 is just a rough figure, not an exact average - you could use 2 or 3. The point of this exercise is to come up with an interest rate that will give a reasonable estimate of the answer.

 

[ssaini]

Thanks Calum for the reply.

Im still a little unsure. I think the main thing i can't get my head around is why t=2.5 i appreciate that its an average i.e. half the time period which is 5. But how come this allows us to estimate the interest over the whole 5 years when there still would be more interest being gained over the time period from 2.5 to 5 which hasn't been considered. Although this gives the correct estimation for the check i can't seem to get my head around it as a concept.

I hope I haven't over complicated things and hope you can help clear it up.

 

[didster]

Basicially you just want an average interest rate (single value) instead of a variable one. So the rate at 2.5 might make a suitable average, or 2 since it's higher at the start, or even 3 might be acceptable.
Once you have this rate you apply it throughout, (over the whole 5 years in this case).

A reasonableness check is just a simple way (ie cut out the complexities) to get the right ball park for the figure.
You can and already did it properly, so no need to over complicate the check.

Soon enough you'll get a feel for what approximations make a big difference and thus how free-handed you could be in pulling average figures from a hat.

 

[ssaini]

Thanks alot for your help guys ive understood it better now.

 

[skhurana]

Hi there,
I think I understand the concept of why the average has been taken but i am little confused on the calculation if you can help with.

When we integrate exp(0.08+0.02/(t+1)) the + sign in the middle changes to X in the answer as e^(0.08(t2-t1)) X ((t2-1)/(t1-1))^0.02. I think I am missing a very basic concept here but If you can clarity that would be great.

I did my A levels 8 yrs back so struggling with some little things.

Kind Regards

 

[John Lee]

Hi there,
I think I understand the concept of why the average has been taken but i am little confused on the calculation if you can help with.

When we integrate exp(0.08+0.02/(t+1)) the + sign in the middle changes to X in the answer as e^(0.08(t2-t1)) X ((t2-1)/(t1-1))^0.02. I think I am missing a very basic concept here but If you can clarity that would be great.

I did my A levels 8 yrs back so struggling with some little things.

Kind Regards

We've made use of the following rules:

exp(a+b) = exp(a) × exp(b)

ln(a^b) = b×ln a

exp (ln a) = a

So:

exp(0.08t + 0.02ln(t+1)) = exp(0.08t) × exp(0.02ln(t+1))

but 0.02ln(t+1) = ln((t+1)^0.02)

and exp{ln((t+1)^0.02)} = ((t+1)^0.02)

Hence:

exp(0.08t + 0.02ln(t+1)) = exp(0.08t) × ((t+1)^0.02)

 

[skhurana]

Thanks a lot.

It makes more sense now

Kind Regards

Sarika