Discrete Compounding vs. Continuous Compounding: An Overview
People invest with the expectation of receiving more than what they invested. That added amount is commonly referred to as interest. Depending on the investment, interest can compound differently. The most common ways interest accrues is through discrete compounding and continuous compounding.
Discrete compounding and continuous compounding are closely related terms. Discretely compounded interest is calculated and added to the principal at specific intervals (e.g., annually, monthly, or weekly). Continuous compounding uses a natural log-based formula to calculate and add back accrued interest at the smallest possible intervals.
Interest can be compounded discretely at many different time intervals. Discrete compounding explicitly defines the number of and the distance between compounding periods. For example, an interest that compounds on the first day of every month is discrete.
There is only one way to perform continuous compounding—continuously. The distance between compounding periods is so small (smaller than even nanoseconds) that it is mathematically equal to zero.
Key Takeaways
- Compounding occurs when interest is paid not only on account balances but on previously-paid sums of interest.
- This "interest on interest" can lead to increasingly large returns over time, and has been heralded as the "miracle" or "magic" of compound interest.
- How often interest is paid on interest matters, as the more often it is paid, the more it will generate over time.
- Discrete compounding refers to payments made on balances at regular intervals such as weekly, monthly, or yearly.
- Continuous compounding yields the largest net return and computes (using calculus) interest paid hypothetically at every moment in time.
Discrete Compounding
If the interest rate is simple (no compounding takes place), then the future value of any investment can be written as:
FV=P(1+mr)mtwhere:FV=FuturevalueP=Principal(r/m)=Interestratemt=Timeperiod
Compounding interest calculates interest on the principal and accrued interest. When interest is compounded discretely, its formula is:
FV=P(1+mr)mtwhere:t=Thetermofthecontract(inyears)m=Thenumberofcompoundingperiodsperyear
Continuous Compounding
Continuous compounding introduces the concept of the natural logarithm. This is the constant rate of growth for all naturally growing processes. It's a figure that developed out of physics.
The natural log is typically represented by the letter e. To calculate continuous compounding for an interest-generating contract, the formula needs to be written as:
FV=P×ert
Many credit cards compound daily, resulting in extremely high credit card balances that are difficult to pay off. Make sure you're aware of how your credit card calculates interest and aim to pay off your balance every month to avoid increasing levels of debt.
Special Considerations
Interest rates impact people in different ways. For example, an investor wants to earn the most interest possible, as it brings more of a return to their initial investment amount. A borrower wants the least amount of interest possible, because that makes the cost of borrowing lower. More interest on borrowed money ends up making it more expensive.
As an individual, you want to ensure that you are finding the best interest profile for yourself. In the case of an investor, they would benefit from compounding rather than simple interest, because simple interest calculates interest only on the principal amount. While this is good for borrowers, it is bad for investors.
As an investor, compounding is always the best scenario; however, if you can receive continuous compounding over discrete compounding, that is even better.
What Is an Example of Compounding Interest?
Compounding interest is interest earned on interest. For example, say you invest $5,000 that earns 5% every year. After the first year, you would have earned $250. This would be added to your initial investment of $5,000, for a new balance of $5,250. In the second year, you would earn 5% not on the $5,000, but on the $5,250. So in year two, you would've earned $262.50 in interest. Your new balance amount would be $5,512.50. In year three, interest would be calculated on the new balance of $5,512.50. You can see how with compounding interest you earn more interest over time.
What Are the Different Amounts of Time Interest Can Be Compounded?
Interest can be compounded at any time. It is dependent on who is determining the compounding intervals. Interest can be compounded daily, weekly, monthly, or annually. The more often it is compounded, the more interest is earned, and the faster your money grows.
Is Simple Interest Good?
Simple interest is good for a borrower because it calculates interest on just the principal amount, not on the interest amount that has accrued, making the cost of borrowing lower for a borrower. Simple interest is not good for an investor, as interest is only earned on the principal amount, but not on the accumulated interest, which would earn more money faster.
The Bottom Line
Compounding refers to how interest is calculated on interest on an investment. The two most common methods, discrete compounding and continuous compounding, will have different outcomes on the return of an investment.
Continuous compounding adds more interest, so it is better for investors, whereas discrete compounding adds less. However, all forms of compounding are better for investors than simple interest, which only calculates interest on the principal amount.
