Continuous compounding only makes big difference at nominally 5+ %

Question

I tried to get a sense of how much difference is makes when a nominal annual interest rate is capitalized multiple times per year, especially as the number of capitalizations approaches infinity, i.e., continuous compounding. I calculated effective annual interest rates for nominal rates of 1%, 3%, 7%, and 10%. The number of capitalizations / year were 1, 2, 4, 12, 50, and 100. 100 is when the curve flattens out, so it's a good proxy for continuous compounding (infinite capitalizations / year).

It seems that multiple capitalizations / year only starts to make a significant difference for nominal interest rates of 5+ % / year. Does this look reasonable?

Matlab/Octave code

cCapPrYr = [ 1 2 4 12 50 100 ]'; % Column of capitalization periods/year
rNomInt100 = 1:3:10; % Row of nominal interest [%]
EffInt100 = zeros( length( cCapPrYr ), 0 );
for NomInt = rNomInt100 / 100
   EffInt100 = [ EffInt100 100*( ...
      ( 1 + NomInt ./ cCapPrYr ) .^ cCapPrYr - 1
   ) ];
end % for NomInt100

plot( cCapPrYr, EffInt100 )
xlabel('Capitalization periods / year')
ylabel('Effective interest / year [%]')
legend('1% nominal','3% nominal','7% nominal','10% nominal')

EffInt100
%    1.0000    4.0000    7.0000   10.0000
%    1.0025    4.0400    7.1225   10.2500
%    1.0038    4.0604    7.1859   10.3813
%    1.0046    4.0742    7.2290   10.4713
%    1.0049    4.0794    7.2456   10.5061
%    1.0050    4.0802    7.2482   10.5116.
%
% Imported into LibreOffice Calc, printed PDF must be cropped:
% pdfjam --keepinfo  --trim "2.5in 8.3in 1.5in 1in" --fitpaper true \
%  EffIntVsCapsPrYr.pdf --outfile EffIntVsCapsPrYr-jam.pdf

Answer 1

Your math checks out.

In practice, however, 'continuous' compounding is basically used for theoretical purposes only, and only as a shorthand for financial planners and the like. (The math involved in continuous compound interest is easier than compounding per second).

Since daily compounding is basically indistinguishable from continuous compounding for any plausible rate of return, it's sufficient - and comparatively simple to implement, thus reducing transaction costs - for virtually all purposes. While fewer compounding periods would be advantageous for financial institutions, competition ensures that they will offer whatever rates/terms bring customers through the door until the cost of acquiring new customers exceeds the benefit of doing so.

Since banks are regularly closing their books on a daily cycle anyway, daily compounding aligns with existing practices very neatly. It's basically a Goldilocks solution that balances the effects of the math you illustrate with the interests of banks.

Answer 2

The discrepancy comes from how the interest rate for the sub period is calculated. If your annual interest rate is 6% most people would calculate the monthly rate as 6%/12 = 0.5%. Mathematical that's actual wrong. The correct "uncompounded" rate would be (1+0.06)^(1/12)-1, i.e. the twelfth root of 1.06 minus 1 which comes out to be 0.48676% and if you use that there will be no difference between compounding or not.

For small percentage rates, the difference between the two methods is quite small but I'm actually not sure what the banks typically do

Answer 3

Re. "Does this look reasonable?"

Various expressions of interest should not produce different results if used properly. Nominal interest rates for various compounding periods can be calculated from an effective rate, as shown in the table below. They should state their compounding interval when quoted. For example, 3 one year calculations using the highlighted rates below produce consistent returns.

$100 at 3.92849% nominal compounded monthly

$100*(1 + 0.0392849/12)^12 = $104

$100 at 7% nominal compounded once per annum

$100*(1 + 0.07/1)^1 = $107

$100 at 9.53102% nominal compounded continuously

$100*e^0.0953102 = $110

The examples above produce the expected returns for 4%, 7% & 10% effective pa.

More examples

Applying continuous compounding for, say, five years at 10% effective pa uses the well-known A = e^(rt) formula, e.g.

$100*e^(0.0953102*5) = $161.051

and the same result is produced with 9.54011% nominal compounded at 50 periods per annum

$100*(1 + 0.0954011/50)^(50*5) = $161.051

Nominal rates may not usually be quoted as such odd decimals above, e.g. one would more likely see "10% nominal compounded monthly", but then that does not return 10% annual effective, nor is it the same as 10% nominal compounded daily.