Continuous compounding at an interest rate of 100% is unlikely to be used in practice. More generally, if the interest rate is x% per annum and interest is compounded n times during the year (so that at the end of each sub-interval,
the amount increases by a factor of (1 + (x/100)/n) ), then the amount has
increased over the year by a factor of
(1 + (x/100)/n)^n which is approximately e^(x/100) = 1 + (x/100) + (x/100)^2/2 + (x/100)^3/6 + .... when n is large.
Mathematically, e^(x/100) is the limiting value of (1+(x/100)/n)^n as n
tends to infinity.
Heuristically, (1 + (x/100)/n)^n gets closer and closer to e^(x/100)
as n gets larger and
larger (from quarterly to monthly to daily to hourly .... compounding).
So, let us turn the problem around. If the APR is specified as y% per annum,
then let x be the solution to the equation
e^(x/100) - 1 = (y/100)
which gives x = 100 log_e (1 + y/100)% as the rate that would be quoted
for continuous compounding while the rate for monthly compounding would
be quoted as the solution to
(1 + (x/100)/12)^12 = 1 + y/100
which gives x = 12 times 100 times the 12th root of (1 + y/100) %.
As a comparison, an APR of 5% per annum corresponds to a
quoted rate of 4.88894...% per annum compounded monthly and 4.8790...% per
annum compounded continuously. Weekly and daily compounding would
result in quotes somewhere in between these two figures, but as
you can see, for a given APR, continuous compounding really does
not make that the quoted rate significantly smaller than the more
common monthly compounding used for mortgages, auto loans, and