Even the most financially illiterate among us can likely explain what an interest rate is: simply, it's what the borrower has to pay for the privilege of borrowing money, or what a saver earns in return for making an investment. Banks and lenders determine the interest rate they apply to consumers in both directions. These rates are widely publicized with terms such as "APR" and "APY," terms which sound similar but actually make a significant difference to the amount of cash you earn or pay.

## What Is Compound Interest?

Imagine you put $1,000 into a savings account. The deposit earns 5 percent interest, so after one year you have earned $50. With simple interest, the initial $100 deposit keeps earning the same 5 percent rate year after year. In year two, you will earn another $50, and again in year three. After three years, you will have earned a $150 return on your investment, which means you have $1,150 in total.

With compound interest, the interest you earn in each period is re-injected into the initial capital, and the interest is then applied to the bigger pot – in other words, **you earn interest on the interest.** In this example, you still will earn $50 in year one, but in year two, you earn 5 percent of $1,050 or $52.50. In year three, you earn 5 percent of $1,102.50 or $55.13. After three years, you have made a return of $157.63, which mean you have $1,157.63 in total – $7.63 more than if your savings had earned simple interest.

The strength of compound interest is in the time you wait: the longer you leave an investment, the more powerful the compounding effect. There's a riddle that illustrates the point perfectly. On a lake, a water lily doubles in size every day, and after 100 days, it covers the entire surface of the lake. What day does it cover half of the lake? Answer: On the 99th day.

## What Is APR?

Before we get to the issue of daily versus quarterly compounding, it's important to understand how jargon like "APR" and "APY" can impact the amount of cash your investment is actually earning, or how much interest you are liable to pay on a loan.

APR stands for annual percentage rate. It describes the **exact interest** your savings will earn (or your loan will accrue) in a year without taking compounding into account. So, if a credit card company charges 1 percent interest each month, the APR would be 1 percent multiplied by 12 months, or 12 percent per year.

## What Is APY?

APY stands for annual percentage yield. This rate does take into account the effect of compounding, assuming that you leave cash in the investment vehicle for a full 365 days. If you like doing math, here's the formula for calculating APY:

**APY = (1 + r/n )n - 1**

Where:

- r is the annual interest rate
- n is the number of compounding periods per year.

The "n" figure here is important. Interest may be compounded on all sorts of time frequencies – daily (365 times a year), monthly (every calendar month or 12 times a year), quarterly (every three months or four times a year), semi-annually (every six months or twice per year) or annually (once a year).

You can hopefully see from the formula how the frequency of compounding potentially can make a massive difference to the amount of interest your savings will accrue.

## How the Daily Compounding Definition Works

Drilling down into daily versus quarterly compounding, the easiest way to see the impact of different intervals is to look at a couple of compound interest examples. An investment that compounds daily adds interest to your account balance every single day, 365 days of the year.

**Example:**

Consider a $250,000 mortgage loan with a 10 percent interest rate accrued daily. Assuming the "year" for this product is 365 days (some banks use 360 days), then the daily interest rate is 10 percent divided by 365, or 0.0274 percent. This figure has been rounded to four decimal places for convenience.

On day one, the interest of the mortgage is equal to $250,000 x 0.0274 percent (0.000274), or $68.50. This is the daily accrual amount. However, because interest is compounding daily, then every day is a "compound date" where the accrued interest is summed and becomes the new base balance.

In other words, the account balance at the beginning of day two equals $250,068.50. Now, the interest is calculated as $250,068.50 x 0.0274 percent or $68.52. By the close of business on day three, the account balance is $250,137.02, and so on.

## Interest Grows as the Lily Unfolds

Remember the water lily example? So far, the changes in the daily interest rate may look fairly small. But if we fast-forward 10 years, you get a much clearer picture of how daily compounding might impact the account balance.

Below is the formula for calculating compound interest with any compounding frequency. We're going to run the calculation longhand for the sake of illustration, but if you search online, there are plenty of compound interest calculators that can do the heavy lifting for you.

**A = P (1 + r/n)** **^** **(nt)**

Where:

- A is the future value of the investment or loan, after compound interest is applied
- P is the principal deposit or loan amount
- r is the annual interest rate (expressed as a decimal)
- n is the number of times that interest is compounded per year
- t is the number of years the money is invested or borrowed for
- ^ means "to the power of"

So, after 10 years, we have:

A = $250,000 (1+0.1/365) ^ (365 x 10)

A= $250,000 (1.000274) ^ (3650)

A = $250,000 (2.718)

A= $679,500 (rounded)

After 10 years, you will owe **$679,500**, assuming you have not paid any of the principal down.

## A Look at Quarterly Compounding

With quarterly compounding, the lender will calculate interest on your account just once every three months, not every day, so the numbers will look different. Using the previous $250,000 mortgage loan example, the initial daily accrual amount will be just the same as with daily compounding: $68.50. Only with quarterly compounding, the accrual amount stays the same for each day in the quarter.

On the compound date, all of those $68.50 sums are added together to form a new base amount. Since a quarter is approximately 90 days, this means that $6,165 ($68.50 x 90) is added to the loan's base balance. In quarter two, you will be paying a daily interest rate of $256,165 x 0.0274 percent, or $70.19 every day until the next quarter day.

What does quarterly compounding looks like after 10 years? Let's run the compound interest formula again, on the basis of four compounding periods per year:

A = P (1 + r/n) ^ (nt)

A = $250,000 (1+0.1/4) ^ (4 x 10)

A= $250,000 (1.025) ^ (40)

A = $250,000 (2.685)

A= $671,250 (rounded).

After 10 years, with daily compounding, you will owe **$679,500**, and with quarterly compounding, you will owe **$671,250**, a savings of **$8,250**.

## Which Compounding Frequency Is Better?

As to which compounding frequency is better, it depends. Generally, **borrowers are better off with less frequent compounding periods**, while **savers are better off with more frequent compounding periods**.

When you are comparing financial products, bear in mind the APY (compounded interest) of an investment will always be higher than the APR (simple interest) of that same account. Banks and financial institutions have a sneaky way of choosing the acronym that makes the rate look as low as possible (for loans) or as high as possible (for investments), so it's helpful to know what the rate you're being quoted actually means.

If you are comparing APYs, take a hard look at how often the compounding occurs and make sure you are comparing products with like-for-like compounding intervals.

### References

- Certificate Calculator | Navy Federal Credit Union
- Bank Tracker: Compounding Interest Daily vs. Monthly: What’s Better for Your Savings?
- The Calculator Site: Compound Interest Formula
- The Calculator Site: Compound Interest Calculator
- Compound Interest Calculator | Investor.gov
- CreditKarma: What Is Compound Interest?