Chapter 14: Instantaneous Forward Rates

[Gbob1]

Hi all, me again

I am referring to page 10 of chapter 14. For the mathematical proof of Ft I am stuck on the fourth line. How does:

-lim(r->0)[logP(t+r) - logPt]/r = -d/dt*logPt?

From what I remember from Maths A-level we'd only just about touched upon the use of lim. But I can't quite remember what it means and what it does. Does lim(r->0) mean the limit as r tends to 0? i think it does but am not sure.

Also it then shows that -d/dt*logPt = -1/Pt*d/dt*Pt

How do I get to this?

And finally how does -1/Pt*d/dt*Pt --> Pt = e^-(integral of Fs with boundaries t and 0)ds ?

Thank you.

 

[Calum]

Disclaimer: I only got taught this yesterday, so I may well be making a hash of it!

That expression involving limits is actually the mathematical definition of differentiation[1]. You've got the right idea, yes. You're asking what happens as h becomes arbitrarily small (but not actually equal to zero). So to answer your question, you just have to recognise that is a definition and apply it.

Differentiating log(Pt) - Pt is a function itself, of t, so you need the chain rule:

d/dx f(g(x))=g'(x)f'(g(x)). Differentiating log(Pt) gives 1/Pt times Pt differentiated.

To answer your last question, you need to step back and think about what F represents. In the discrete time case, F would be the interest rate agreed now to apply at some time period in the future, between t and t+r, and the accumulation factor is simply Pt/Pt+r. In the continuous case, we have a similar idea, but since we have a force of interest we have exp(F(t,t+r))=Pt/Pt+r. If we let r tend to zero, we get the "instantaneous forward rate", the force of interest for a very small time period. To finally answer your question, then, we have the instantaneous rate for any instant of time, and therefore to find Pt all we have to do is integrate to sum up all these instant rates.

[1] More generally, f'(x)=limit(h->0) (f(x+h)-f(x))/h. Try it with something simple like f(x)=x^2.

 

[John Lee]

That expression involving limits is actually the mathematical definition of differentiation.

He's right - but again from the exam point of view you just have to be able to use the (1+f) = e^F rule to calculate it.

This is the frustration with the CT1 Core Reading - they cover the theory but the Profession's exams (unlike many Uni's exams on this) are actually very numerical calculation based. So it's easy to get hung up on the theory rather than the application.

I believe the reasoning behind this is that you need to be able to use these results in the workplace (and have an appreciation of why they work) rather than prove them to prove a higher education point!!!

So only proofs you'll need from CT1 are the annuity and increasing annuity formulae in Chapters 6 and 7, the derivations of the mean and variance formulae in Chapter 15 and the proof of the equivalence of volatility - vDMT in Chapter 14.

Hope that helps you cut your workload a bit!

 

[Gbob1]

He's right - but again from the exam point of view you just have to be able to use the (1+f) = e^F rule to calculate it.

This is the frustration with the CT1 Core Reading - they cover the theory but the Profession's exams (unlike many Uni's exams on this) are actually very numerical calculation based. So it's easy to get hung up on the theory rather than the application.

I believe the reasoning behind this is that you need to be able to use these results in the workplace (and have an appreciation of why they work) rather than prove them to prove a higher education point!!!

So only proofs you'll need from CT1 are the annuity and increasing annuity formulae in Chapters 6 and 7, the derivations of the mean and variance formulae in Chapter 15 and the proof of the equivalence of volatility - vDMT in Chapter 14.

Hope that helps you cut your workload a bit!

Very helpful once again. I've taken a break from studying for a few days cos I'm up in Norwich but had a brief look at chapter 15 last night and boy does it look complex. I swear I've never even learnt half the stuff - or maybe I did and I didn't listen. I do vaguely remember variance and standard deviation, but not the way they're teaching it. Will have to dig out the statistics book at some point!