 What is compound interest? - InterestRate.co.uk # What is compound interest?

The world of personal finance can be confusing for many. You might come across various financial products and services, from mutual funds and money market accounts to student loans and the stock market.

When looking at some financial products from lenders, you will come across many terms such as annual percentage rate, accumulated interest, APR, APY, and more. You will also hear the terms simple interest and compound interest. To understand the basics of compound interest, we need to know how simple interest works.

## Explaining simple interest and how compound interest works

So, let’s look at simple interest and how the formula for compound interest works. For instance, take a savings account that offers 10% interest per annum. This could be a simple interest or compound interest account. The basics are as follows.

### Simple interest

If you held an amount of money, such as £1,000, in a deposit account for one year, then, using the above figure, you would receive £100 in interest on the 12-month anniversary. If you were to withdraw the interest each year, all things being equal, your savings would earn £100 per annum.

### Compound interest

Suppose you held £1,000 in a deposit account for one year, which paid interest (kept in your account) every six months. In this scenario, you would receive an interest payment after six months and then 12 months as you move into the second year. Your full interest payment for the 12 months would be £102.50. The difference is easily explained, as you would receive interest on the interest credited to your account after six months based on the frequency of compounding. This is the power of compound interest and exponential growth:

Interest on £1,000 for first 6 months (6/12ths of annual rate) = 5% x £1,000 = £50

Interest on an increased balance of £1,050 (includes first interest payment) from 6 months to 12 months (6/12ths of annual rate) = 5% x £1,000 = £52.50

In this example, the difference between compound interest and simple interest received is £2.50. This is the final amount of six months of interest on the initial £50 of interest:

Interest on £50 for 6 months (6/12ths of annual rate) = 5% x £50 = £2.50

We have simplified the above scenarios. You may take funds out of your savings account or add additional funds throughout the year. However, you would not receive 12 months of interest on funds not held in your savings account for the whole period. There would be an element of apportionment relative to the length of time funds were held on deposit.

This is a basic comparison between simple interest and the compound interest formula. We will now look at how to calculate compound interest and frequency and look closer at the power of compound interest and the annual percentage yield.

## Calculating compound interest

When calculating compound interest, there are three variables needed, which are:

• Principal (P)
• Nominal annual interest rate (i)
• Number of compounding periods in full years (n)

To simplify the calculation, we will assume a principal of £1,000, a 10% nominal interest rate compounded over three years.

These variables were used to create two similar formulas:

## Future balance, including compound interest

The formula for this is as follows:

Using the above example, the specific calculation is:

This equates to:

### Compound interest figure

When calculating compound interest in isolation, it is merely a case of using the above formula and then subtracting the initial amount:

Therefore, the calculation is as follows:

In this instance, the compound interest equates to £331 over three years. If the interest had been withdrawn at the end of each 12 months, this would only have totalled £300. Because of compound interest, an additional £31 would have been added if the funds and interest remained in the account for the entire three years.

Related article:   Bank offers free internet to customers struggling with interest rate rises

## The growth effect of compound interest

When looking at the above figures in isolation, it is easy to discount an additional £31 over three years as irrelevant. But what do these figures look like for five, ten, 20 and 30 years?

To calculate the simple interest figure, we have assumed the interest is withdrawn every 12 months – as soon as it is credited. When calculating the compound interest figure, we have assumed the interest payments are not withdrawn.

In this example, the effect of compounding increases the principal by a factor of 1.593742 every ten years. Therefore, we can calculate that after 50 years, the balance, including compound interest, would be equivalent to 117.39 times the original principal.

While compound interest may seem minor over a brief period, it can be significant over ten, 20 or 30 years. The reason is simple: interest on interest. The initial amount of £100 interest in year one would have earned interest for 29 years, compound interest from year two would have earned interest for 28 years, and so on.

## Compounding at different frequencies

The above examples perfectly illustrate the value of compound interest on an annual basis over many years. However, some savings accounts will pay interest monthly, quarterly, or semi-annually. You will also find that some credit cards will add interest to your account daily using compound interest. The frequency at which compound interest is added to your balance is the key to understanding the long-term increase compared to simple interest.

We will now look at various frequencies to show the impact of adding interest several times over a specific period. The following variables will be used for these compound interest examples:

Principal amount: £1 million

Interest rate: 10% per annum

### Annual compound interest

The annual compound interest calculation on a month-by-month basis is as follows:

While the annual interest rate stays constant in this example, at 10%, the interest earned increased from £100,000 in year one means you’ll earn 10% interest on £110,000 in year two. This is the impact of compound interest, as the previous year’s interest adds to the starting balance for year two.

If interest had been withdrawn annually, the total would have been £200,000 against £210,000 in this instance.

A continuation of this table over five years would generate interest of £610,510 instead of £500,000 if simple annual interest were withdrawn each year.

### Quarterly compound interest

The quarterly compound interest calculation on a month-by-month basis is as follows:

There are two figures of interest in these tables. Firstly, the impact of adding interest quarterly, leading to interest on interest, increases the interest rate from 10% to 10.38% per annum. You will notice that interest in year one was £103,813, rising to £114,590 in year two. If interest were paid annually and withdrawn each year, this would total £200,000 against £218,403 on a compound basis.

Related article:   Bank profits soar on the back of rate rises while consumers struggle

If we continue this table over five years, interest earned would increase to £638,616 against £500,000 if simple annual interest had been withdrawn each year.

### Monthly compound interest

The monthly compound interest calculation on a month-by-month basis is as follows:

In this example, the impact of adding interest monthly increases the annual interest rate from 10% to 10.47%. The effect of compounding the interest means that in year one, there was an interest payment of £104,713, increasing to £115,678 in year two. The compound interest over two years was £220,391 compared to £200,000 if annual interest had been withdrawn upon receipt. This equates to over a 10% increase in interest received.

The interest figure increases even further over five years, equating to £645,308 against £500,000 if simple annual interest payments had been withdrawn from the account each year.

## The Rule of 72

When it comes to compound interest, there is an interesting formula called the “Rule of 72”. By dividing 72 by the rate of interest used to calculate compound interest, you learn roughly how long the initial principal will take to double in value. For example:

• 3% interest rate

The basic calculation would be:

72/3 = 24 years to double

• 5% interest rate

The basic calculation would be:

72/5 = 14.4 years to double

• 10% interest rate

The basic calculation would be:

72/10 = 7.2 years to double

• 20% interest rate

The basic calculation would be:

72/20 = 3.6 years to double

This approximate calculation is demonstrated by the formula 〖1.20〗^4 = 2.0736

Using the “Rule of 72,” you can quickly calculate the years it would take for a deposit to double using a specific interest rate. This can be extremely useful when looking at savings accounts and loan arrangements.

## Calculating compound interest on varying frequencies

We have covered the annual, quarterly, and monthly compound interest frequencies, demonstrating the power of compound interest. However, some credit card companies add interest daily. This prompts the question, what interest rate do we use when calculating different frequencies?

### How daily compound interest works

As the interest rate is an annual figure when adding interest each day, this should be based on 1/365th of the annual interest rate or 0.003 in decimal terms. For example, if the interest rate were 10% per annum, the daily interest rate would be:

Related article:   Warning of recession from BoE following another interest rate hike

10% x 1/365 = 0.0274%

So, interest will be calculated and added to the account balance daily at a rate of 0.0274%. The calculated compound return on £1,000, using daily interest over a full year, would be:

### How weekly compound interest works

To calculate the correct interest, we need to divide the annual interest rate by 52 to give the apportioned weekly compound interest rate, i.e., 1/52nd of the yearly rate. Again, using the 10% per annum interest rate, the weekly interest rate would be:

10% x 1/52 = 0.192%

The weekly compound interest figure would be calculated by multiplying the weekly balance by 0.192%. This figure would then be added to the total amount of the balance upon which interest for the next month would be calculated. The short formula for the full year is as follows:

As you can see, the annual interest figure is similar when calculated daily or weekly. If you were to replicate the above formulas over ten years, the figures are as follows:

Compound interest calculated daily = £1718.18

Compound interest calculated weekly = £1711.34

If we used a £100,000 or a £1 million deposit, the interest figures would be more significant and the difference more exaggerated. So, why would a credit card company consider daily interest on small credit card balances?

### Credit card companies, daily interest and changing balances

If you have credit card debt, you are likely charged interest monthly. However, is the interest on your credit card debt calculated monthly?

Upon digging a little deeper into the terms and conditions of your credit card, you will find that interest is charged daily but credited to your account monthly. Note interest is charged on your unpaid balance, not on new transactions and withdrawals, which are paid at the end of each billing cycle. So while monthly compound interest will increase the simple annual interest rate, daily compound interest will increase this figure even more. So how does this work?

In the following examples, we will assume that purchases during the month are not repaid at the end of the billing cycle. Consequently, you tend to find that interest is then backdated to the date of each purchase.

The credit card company will charge 1/365th (in reality, some may work on a 360-day year, i.e., 1/360th) of their annual interest rate every day on your account balance. This interest will be added to your account daily under the compound interest rate formula.

While many people feel as though they have an element of “interest-free” funding, if they spend on their credit card at the start of the month, this is not the case. If you spent £1,000 at the beginning or the end of the month, you would see a difference in the amount of interest charged. A simple annual interest rate of 10% on your credit card equates to the following over different frequencies:

• 10.52% if the compound interest is calculated daily
• 10.51% if the compound interest is calculated weekly
• 10.47% if the compound interest is calculated monthly

The typical credit card tends to have a simple interest rate of around 20%, which equates to the following compound interest rate figures on varying frequencies:

• 22.13% if the compound interest is calculated daily
• 22.09% if the compound interest is calculated weekly
• 21.94% if the compound interest is calculated monthly

If you have a low credit rating, you may be charged more on your credit card, possibly 30% per annum or above. The compound interest rates for a credit card which charges 30% a year would be as follows:

• 34.97% if the compound interest is calculated daily
• 34.87% if the compound interest is calculated weekly
• 34.49% if the compound interest is calculated monthly

When you look at credit card terms and conditions and rates, you will often see two rates quoted: the simple rate and the compound rate. Credit card companies are legally obliged to list the compound interest rate in their promotional literature as this is the “real” annual interest charge. However, it is essential to distinguish the compound interest rate from the annual percentage rate (APR), which relates to simple interest and the impact of additional charges on repayments.

## The Bottom line

Compound interest is a powerful tool that consumers frequently overlook as “irrelevant” when looking at the amount of interest being charged. We tend to focus on the lower figure, which would be the simple interest rate. Instead, you should always focus on the compound interest rate, which is the “real rate.”

When looking at the impact of compound interest, there are four main factors to take into consideration, which are:

• Rate
• Balance
• Term
• Frequency

The more often interest is charged or added, for example, daily, the greater the element of interest on interest going forward. The larger the balance, the larger the interest figure in monetary terms, and the longer the term, the more significant the impact of compound interest. As you can see from our figures, this has an effect on an annual basis but can have a potentially massive impact on long-term arrangements.