Hi all Apologies for asking what is probably quite a simple question. My excuse is that I'm home studying so I don't have any other access to any actuaries, maths gurus or tutors. It's also been over 10 years since maths a-level! I want to check whether I have properly understood how delta t is arrived at - and get some help on the last manipulation. I'll run through it in bits: 1. F(t + h) - F(t) / h This is the rate of change of the value of the fund - F (the y axis) with respect to time (x axis). Putting this in the limit of h (the small increase in time) tending towards 0 is the same as taking the derivative or writing F'(t). 2. The above is divided by F(t) since we want to know the rate of change as a proportion of where it was a moment ago. Like a percentage - dividing the increase by the starting value. So that gets me to F'(t)/F(t) As I understand it, all we've done is rewrite our original formula - we haven't actually differentiated anything? Have I understood this correctly - or have I got to the right place but using faulty logic? Now how do we get to d/dt ln F(t)? I understand that d/dt ln(t) = 1/t, but other than that I'm having trouble seeing it. I suspect that I could just be being a bit dense! Any help appreciated Cheers Rob
Differentiating ln(F(x)) wrt x gives F'(x)/F(x) using the chain rule. Search for "chain rule" in wikipedia or an A'level maths book.
Ok, found it - though if I'm honest I'm not sure how to derive that result - I'm sure it doesn't matter. Thanks for the pointer
u = F(x) y = ln(F(x)) = ln(u) dy/dx = dy/du * du/dx = 1/u * F'(x) = F'(x)/F(x) (replacing u with F(x)) No need to know the proof of chain rule if that's what you meant.
