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Why are we compounding continuously in finance? I have searched around, but I cannot find an explination on why we actually do it.
I assume that we, in theory, do it because every interest earned is reinvested. But since that is not a reality, then what is the answer?
Does it has something to do with the present value changing?
Continuous-time formulation is much easier for some basic asset pricing theories. In continuous-time you will have to deal with integrals rather than sums which makes your life much easier. And for those, you will need continuous discounting. Here's an excerpt from John Cochrane's Asset Pricing:
The choice of discrete vs. continuous time is one of modeling convenience. The richness of the theory of continuous time processes often allows one to obtain analytical results that would be unavailable in discrete time.
Sometimes it is easier to work with continuos compounding in some models, especialy when you compound interest daily. Moreover, it can come from history when it was more difficult to calculate higher powers than tabulated exponential function.
Take for example annual interest rate $i = 5 \%$. Then effective annual interes rate based on daily compounding is $$ \Big(1+\frac{0.05}{365}\Big)^{365} -1 = 5.1267\% \approx 5.127\% $$
If you use continuous compounding you get
$$ \mathrm{e}^{0.05} -1 = 5.1271 \% \approx 5.127\%. $$
As you can see the difference is only 0.0004 percentage points, a negligible value.
