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Periodically and Continuously Compounded Interest
-------------------------------------------------
FV = PV(1 + i)^{Y}
where Y = number of years.
i = stated (nominal) interest rate.
For example, an annual rate of 10% for 1 year would give:
FV = PV(1 + .10)^{1} = 1.10PV
If, instead, the interest rate is compounded on a daily, monthly or
quaterly basis, then the formula becomes.
FV = PV(1 + r/n)^{nY}
where n = number of compounding periods per year.
For example, a stated annual rate of 10% for 1 year compounded
monthly would give:
FV = PV(1 + .10/12)^{12} = 1.105PV
Compounding using periods which are less than one year gives rise
to an "Effective" annual rate that is higher than the "Stated"
rate.
In the limiting case it is possible to compound the interest
continuously, meaning that your balance grows by a small amount
every instant.
Define a variable m = n/r so n = rm. Therefore,
FV = PV(1 + 1/m)^{rmY}
= PV{(1 + 1/m)^{m}}^{rY}
In the limiting case where the time slice becomes very small, we
can write, by definition,
lim(1 + 1/m)^{m} = e
m->∞
Therefore,
FV = PVe^{rY}
Alternatively, PV = FVexp(-rY)
Verification:
Replace FV with f(t) for the balance at time t (with t measured
in years). So:
f(t) = PVexp(tr)
Taking the derivative
df(t)/dt = rPVexp(tr) = rf(t)
In words, this is saying that "at any instant the balance is
changing at a rate that equals r times the current balance"
which of course is the definition of continuous compounding.
Effective Interest Rate
-----------------------
The effective interest rate, r , is given by:
r = (1 + i/n)^{n} - 1
For example, i = 5% = 0.05 annually, n = 12
r = (1 + 0.05/12)^{12} - 1 = 0.0512 = 5.12%