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 annual percentage
yield (not the same as 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
as the APR for continuous compounding while the APR 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 the APR.
As a comparison, an annual percentage yield of 5% per annum corresponds to a
quoted rate (APR) 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 annual percentage yield, continuous compounding
not make that the APR significantly smaller than the more
common monthly compounding used for mortgages, auto loans, and