Discrete Compounding

What Is Discrete Compounding?

Discrete compounding refers to the method by which interest is calculated and added to the principal at certain set points in time. For example, interest may be compounded weekly, monthly, or yearly.

Discrete compounding can be compared with continuous compounding, which uses a formula to compute interest as if it were being constantly calculated and added to the principal amount.

How Discrete Compounding Works

Compound interest is a process whereby interest is earned in subsequent periods on interest that has already been earned in previous periods. Therefore, if you held a deposit account at a bank that paid 1% interest per year, you would receive $1 if your initial balance was $100, but your second-year interest would be computed based on the new amount of $101 that started year 2 (assuming no additional deposits or withdrawals have been made), which results in $1.01 of interest. One penny more than the year before. Of course, these sums become much more consequential as one's principal amount grows and interest rates rise.

In the case of the bank account, if interest is paid yearly on the account balance, it is a form of discrete compounding, since the interest is calculated at a discrete-time interval of once a year. Other intervals may include monthly, weekly, or daily. Certain loans or credit cards may charge daily compounding interest, which means that your amount owed can quickly grow to very large amounts.

Note that not all interest-bearing instruments feature compounding. Thus, if you own a fixed-rate bond paying 10% annually with a $1,000 face (par) value, you would be paid $100 per year only on the $1,000 face amount.

The future value of an account that has interest compounded discretely can be calculated as follows:

FV = P ( 1 + r m ) m t where: t = The term of the contract (in years) m = The number of compounding periods per year \begin{aligned} &\text{FV} = \text{P} (1+ \frac{r}{m})^{mt}\\ &\textbf{where:}\\ &t = \text{The term of the contract (in years)}\\ &m = \text{The number of compounding periods per year}\\ \end{aligned} FV=P(1+mr)mtwhere:t=The term of the contract (in years)m=The number of compounding periods per year

The Effect of Compounding Frequency

The frequency with which interest is compounded has a slight effect on an investor's annual percentage yield (APY).

For example, suppose you deposit $100 in an account that earns 5% interest annually. If the bank compounds interest annually, you will have $105 at the end of the year. If, on the other hand, the bank compounds interest daily, you will have $105.13 at the end of the year.

Though not technically "continuous" at every second, continuous compounding is considered as such if compounding takes place on a daily basis.

In the simple illustration above, you can see that less frequent compounding means fewer interest earnings in your bank account. Even Wells Fargo, which has shown disrespect to millions of its customers by creating fake accounts to dishonestly pile up bank profits, compounds interest daily. The APY, therefore, is higher than yields under discrete compounding that would take place monthly, semi-annually, or annually.

However, the Wells Fargo customer is not exactly jumping up and down with excitement as of the third quarter of 2020. Interest rates in the economy have been falling, but Wells Fargo's APY in basic checking and savings accounts is even lower at 0.01%. Wells Fargo Way2Save Savings accounts pay 0.01% in interest.

That means if you put $10,000 into the savings account, you would earn a meager $1.00 in interest for the entire year. Your savings account would have a balance of $10,001. That's not exactly a great way to save, but take solace — you can withdraw that one dollar from the bank and go to Starbucks and buy a half cup of coffee.