I recently had learned about the concept of compounding continuously. How does it work? How is compounding something continuously not infinite? What is e and why is it necessary?
In: Economics
e is magic number, like pi. There is an algorithm to get an exact value of the number e^(x) where x can be any number, positive or negative, whole or a fraction, and the algorithm will just keep giving your digits until you get tired of it. Getting an exact power, like 2^(5.122) is possible using this formula, and adjusting it for the fact that 2 is not e.
Compound interest is usually compounded in increments of whenever interest is paid… usually monthly for some banks accounts, perhaps yearly for some bigger investments, and so on. But mathematically, the shorter the period, the better. If you have 12% interest (for the sake of easy math), but compound monthly, each monthly interest payout is only 1% of your bank account’s total. But doing it that way is still worth more than one interest payout of 12% at the end of the year because the interest is compounding month to month.
“Continuous compounding” takes it to the mathematical extreme. Oh, and obviously you need to be able to have tiny fractions of a penny in your bank account to work like this but… What if interest was paid every minute? Every second? Every millisecond? What happens if the interval becomes effectively 0? Now you have continuous compounding, and the extreme limit. You’ll need to use e to get the value.
I’m not aware of anyone actually *doing* this. Few banks want to give their customers more money, ya know?
The more frequently you compound interest, the less you compound each time. Still, more compounding events gives you a bit more money relative to doing it less frequently at the same interest rate. If you calculate the difference between compounding every year, every month, every day, even every second, you’ll notice that each one is higher but eventually the differences become smaller and smaller.
This is because they are converging on a theoretical maximum that you can never get higher than no matter how frequently you compound an increasingly tiny amount.
You can use calculus to calculate this theoretical maximum, and continuously compounding interest means that the person paying you interest is just giving you that theoretical maximum which is usually fractions of a percent more than compounding ever day or week.
Put another way, if you take the expression (1+1/x)^x and plug in numbers for x, you will notice that you will start to approach but never reach 2.718, no matter how large of a number you choose for x. This is e, and since that is the general expression for interest we can substitute e and say it’s equivalent to using the largest possible x, which is the frequency of compounding.
Continuous compounding will have a “rate/time” value, such as 5% per year. This results in a formula something like Result=Initial*rate^(amount of time periods). A similar formula is involved with half lives.
Using that rate/time value, you can substitute a smaller rate and a smaller time period, such as 2.5% per 6 months (which means more time periods). This ends up increasing the Result, but the rate of that increase is smaller as you decrease the size of the time periods.
Calculus lets us calculate what the rate would be if the time periods were infinitely small (aka the interest happens continuously). This involves the number *e*, which is the result of combining exponents and dividing them against each other (which is what paragraph 2 is doing, and is why it shows up here).
You start with apr. that is how much interest you get in one year. Most loans compound monthly so you divide the apr by 12 and that is your monthly interest rate. But say you compound weekly. You divide the apr by 52 (weeks in a year). That is how much interest you get weekly. Daily would be apr divided by 356.
But if you want to compound continuously you can’t divide by infinity. So you get a more complicated equation of Pe^(rt). So you can say apr is r, t is time since the loan started, P is the principal amount, and e is a constant that a very smart mathematician calculated and we accept because it’s just right and we don’t know enough math to disprove. We can use this to calculate the amount of interest at any given point in time.
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