Why Continuous Compounding Is Used in Financial Models
Summary
The document explains that continuous compounding is often a modeling choice rather than an assumption that interest is literally reinvested at every instant. Continuous-time asset-pricing theories describe evolving prices and discounting with continuous processes, which can make their mathematics more tractable and sometimes yield analytical results that are harder to obtain in discrete time.
It also notes that continuous compounding can approximate frequent discrete compounding closely. An example compares daily compounding at a stated annual rate with continuous compounding and finds only a small difference in the resulting effective annual rates. The discussion does not claim that continuous compounding is universally more realistic or preferable; its appeal depends on the model and the convenience of the continuous-time framework.
Key ideas
- Continuous compounding is commonly chosen to fit continuous-time financial models.
- Continuous-time formulations can simplify asset-pricing mathematics and enable analytical results.
- Continuous compounding can closely approximate frequent discrete compounding.
- The choice between discrete and continuous time is a modeling convenience, not a claim about literal reinvestment at every instant.
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Full text
# Why continuously compounding
# Why continuously compounding
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?
## Answer by MarcusAerlius (score 3, accepted)
https://quant.stackexchange.com/a/53052
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.
## Answer by Martin Vesely (score 2)
https://quant.stackexchange.com/a/53048
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.