A common explanation for the mathematical constant e (2.71828...) is that it is the factor by which an investment would grow at 100% interest rate over a period if it is continuously compounded.

In other words, if you were to invest 1 million dollars for one year at a rate of 100%, after one year, the balance would show about 2.71 M$ (or e M$)

However, it is not a case I have ever seen. We often see monthly compounding, sometimes daily, but I've never seen continuous compounding.

Does continuous compounding exist in practice? Is it ever part of a financial product offering? If so, in which cases might it be used?

  • I think the point is for you to adjust the variables in that formula to a real world scenario. – CQM Nov 24 '12 at 20:17

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 really does not make that the APR significantly smaller than the more common monthly compounding used for mortgages, auto loans, and the like.

  • +1 what a beautiful explanation, an upvote and hat tip, Dilip. – JTP - Apologise to Monica Nov 24 '12 at 21:03
  • If you want 5% gain over the year and want to compound using n intervals you will get 105% at the year end by using initial*(interestPerPeriod + 1)^n where interestPerPeriod = (x + 1)^(1/n) - 1 and x = 0.05. That's the maths at least. The APR method appears to be something different. If I didn't get 105% I'd be worried. – Chris Degnen Nov 14 '13 at 19:36
  • @ChrisDegnen You are working things backwards. Normally, interest rates are quoted as x% per annum compounded z-ly where z could be "annual, quarter, month, week, dai, hour" etc. in which case at the end of the n periods comprising a year, you will get (1+x/100n)^n = 1+y/100 and the APR is y%. You are starting with an APR of y% and asking, if the interest is computed n times in the year, what is the interest-per-period (IPP)? When you solve for IPP, you can get the "quoted rate" as (n times IPP %) compounded n-ly). The APR is required by law to be stated on all loans, but what is used is IPP. – Dilip Sarwate Nov 14 '13 at 20:20
  • The cause of this misunderstanding is that in the European Union, APR (annual percentage rate) is the effective annual rate, or annual percentage yield as you put it (5%). The APR you quote (4.879%) is the nominal rate, or logarithmic rate, which is not quoted in the EU. In the EU monthly rates are calculated as (APR/100 + 1)^(1/12) - 1, which in this example would be 0.4074%. Or from log return, e^(0.04879*1/12) - 1 = 0.4074%. – Chris Degnen Apr 15 '14 at 20:22
  • @ChrisDegnen Yes, APR seem to mean different things in the US and the EU. The 5% annual percentage yield (APY) of the US can, as I said, be stated as an APR of 4.88894...% per annum compounded monthly, making the per-month rate to be 4.88894/12 = 0.4074116..%. which is exactly what you get from the EU APR of 5% via (1+APR)^{1/12}-1. – Dilip Sarwate Apr 15 '14 at 20:33

Here are some calculations for an investment which yields a final value of $ e million ($2,718,282) from an initial value of $ 1 m.

The logarithmic or continuously compounded return is given as:-

enter image description here

Vf = 2,718,282
Vi = 1,000,000
rlog = ln(Vf/Vi) = 1.0 = 100 %

This is a logarithmic return, or nominal return (continuously compounded) of 100%.

The effective annual rate can be calculated by:-

enter image description here

where i is the logarithmic or continuously compounded nominal rate.

i = rlog = 1.0 = 100%
r = e^i - 1 = 1.718282 = 171.8282 %

An effective annual return of 171.8282% produces the final value of $ e million.

Of course, the effective return can also be calculated as:

r = Vf/Vi - 1 = 1.718282 = 171.8282 %

Now considering monthly periodic returns

The effective annual rate calculated from a periodically compounded nominal return is:

enter image description here

where n is the number of compounding periods.

Rearranging this formula, and using the previously calculated effective annual rate (which produced $ e million), the annual nominal rate compounded monthly is calculated:

r = 1.718282 = 171.8282 %
n = 12
i = n*((r + 1)^(1/n) - 1) = 1.0428486 = 104.28486 %

Note how this differs from the 100% calculated for the continuously compounded nominal rate. As n increases the periodic nominal rate approaches the continuously compounded nominal rate, as demonstrated by the limit formula:

enter image description here

For example, the nominal rate compounded daily (with n = 365) is 100.137%, which is somewhat closer to 100% than the nominal rate compounded monthly.

From the annual nominal rate compounded monthly, the monthly compounding rate can be found:

m = i/n = 1.0428486/12 = 0.08690405 = 8.690405 %

Checking by compounding for 12 months: (m + 1)^n - 1 = 1.718282

The monthly compounding rate can also be calculated directly from the logarithmic rate, or annual continuously compounded nominal rate:

i = rlog = 1.0 = 100 %
n = 12
m = e^(i/n) - 1 = 8.690405 %    

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