Home > Archive > APL > May 2006 > FORCE OF INTEREST
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| denver 2006-05-04, 7:56 am |
| hello guys,
i have a problem with the compound interest theory a bit. i know and
understand that there is a nominal interest and an effective interest
the thing that i do not understand is the force of interest on what
exactly are it properties and how to apply it and how does it work.
anyone who can help me, thank you.
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| James J. Weinkam 2006-05-07, 6:58 pm |
| denver wrote:
> hello guys,
>
> i have a problem with the compound interest theory a bit. i know and
> understand that there is a nominal interest and an effective interest
> the thing that i do not understand is the force of interest on what
> exactly are it properties and how to apply it and how does it work.
>
> anyone who can help me, thank you.
>
The force of interest is the relative instantaneous time rate of change of the
value of the interest bearing investment. If v(t) is the value of the
investment at time t, the force of interest at time t is v'(t)/v(t).
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| denver 2006-05-07, 6:59 pm |
| thank you.
let say for an example, (1) find the accumulated amount of 200 invested
at a constant 4% force of interest for 2 years.
which formula must i use?
thank you for your help, it means the force of interest is equal to the
derivative over the actual function. when does d=ln(i+1)? and when do
i have to use the integral i.e s=exp[ integral from T2 to T1 d(s) ds?
i know it sounds confusing, it just that i dont know how to write the
signs.
take d to be the force of interest (delta)
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| don@inetpurchasing.com 2006-05-07, 6:59 pm |
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denver wrote:
> hello guys,
>
> i have a problem with the compound interest theory a bit. i know and
> understand that there is a nominal interest and an effective interest
> the thing that i do not understand is the force of interest on what
> exactly are it properties and how to apply it and how does it work.
>
> anyone who can help me, thank you.
'force of interest' is a term i never heard of. congress once
asked j. p. morgan if he knew the 7 wonders of the ancient world.
morgan said no, but he knew the 8th wonder - compound interest. he
was also asked what will the stock market do next year and replied 'it
will fluctuate'.
200 at 4% compounded for 2 years is 200 x 1.04*2 = 216.32.
if you know the end result and want the interest rate: (216.32 divide
200)*divide 2 = 1.04
you can do it with e=(1+divide n)*n with n = infinity for
instantaneous rate = 2.71828.....
(x*2) = 216.32 divide 200 = 1.0816 = the multiplier.
solving for x: x = *(ln 1.0816) divide 2 = 1.04 qed.
don mattern
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| James J. Weinkam 2006-05-07, 6:59 pm |
| denver wrote:
> thank you.
>
> let say for an example, (1) find the accumulated amount of 200 invested
> at a constant 4% force of interest for 2 years.
>
> which formula must i use?
> thank you for your help, it means the force of interest is equal to the
> derivative over the actual function. when does d=ln(i+1)? and when do
> i have to use the integral i.e s=exp[ integral from T2 to T1 d(s) ds?
> i know it sounds confusing, it just that i dont know how to write the
> signs.
> take d to be the force of interest (delta)
>
I'll run through this one for you but in the future you'll have to do your own
homework:
From your stated conditions one has
v(0)=200
v'(t)=.04*v(t)
A general solution of the differential equation has the form v(t)=c*exp(a*t)
Therefore c=200 and a=.04. Thus v(t)=200*exp(.04*t)
For t=2, v=216.6574135~=216.66
This should be compared to 200*(1.04)**2=216.32
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| phil chastney 2006-05-07, 6:59 pm |
| denver wrote:
> hello guys,
>
> i have a problem with the compound interest theory a bit. i know and
> understand that there is a nominal interest and an effective interest
> the thing that i do not understand is the force of interest on what
> exactly are it properties and how to apply it and how does it work.
Denver -- do you understand the motivation for measuring the force of
interest? a nominal rate of 5%, say, compounded half-yearly, gives an
effective rate of interest greater than 5% -- compounded quarterly, it
gives an even higher rate of interest -- so as we decrease the time
interval between compounding, the effective rate of interest increases
or, conversely, you might ask, what nominal rate of interest,
compounded quarterly, gives the same effective annual rate as
5% compounded annually?
so what happens when the compounding interval is zero?
i.e, what if we compound instantaneously?
the force of interest is a useful number to have -- with discrete
compounding intervals, we only know the value of an investment at
those points in time where compounding occurs -- using the force
of interest, we can give a figure for the value of an investment
after 0.31415 years, or whatever you interval you like
you going to need to know this -- after all, claims don't all come
on quarter days, so you will be needing to see returns on
investments as a cash flow -- as a continuous stream, if you like
also, it gives you a measure which enables you to compare the
returns on all sorts of investments -- you will find all sorts of
crappy measures used in the investment field -- obviously, you have
to learn their lingo to talk to these guys, but at the end of the
day, the only reliable measure is force of interest
so, once you see this thing as the continuous limit of ever shorter
discrete compounding intervals, the maths is a piece of cake
HTH . . . /phil
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