*The Business Professor*, updated October 6, 2019, last accessed July 11, 2020, https://thebusinessprofessor.com/lesson/compound-interest-definition/.

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### Compound Interest Definition

Compound interest (financially called compounding interest) is interest accrued on the actual principal plus accumulated interests over a period of time. Simply put, compound interest is the interest on interest of a starting capital. Unlike simple interest, this compounding actually calculates all parts of the equity in offering the new interest. Having originated in 17th century Italy, compounding interest is generally referred to as “interest on interest” and this makes its a better counterpart to simple interest depending on the parties involved in given agreements. Since simple interest typically calculates interest on initial principal even over a period of time, the payout is more likely to be smaller than when same initial principal is given interest rates via compounding. It is important to note that the frequency or rate of accruement in compound interests is the most essential factor in determining payouts. The more the number of compounding periods, the higher the payouts and vice-versa. Simply put, for a $3000 initial principal commanding a 1% interest rate annually for a single year, the compound interest would be $30. If this same sum is commanding a 1% interest monthly for a year, the annual interest payout would be approximately $379. The difference between both interest payouts is due to the “interest on interest rule” used by compounding interests. Here, the first 1% interest would be added to the initial principal to calculate the second 1% and this would go on till the 12th month. This accruement is referred to as the “miracle of compound interest” as some suggest that the upcoming interests are totally investment free.

### A Little More on What is Compound Interest

Compounding interest is measured by multiplying the actual deposit (or principal) amount by one, and then raising the answer to the power of the compounding periods (monthly, semi-annually, annually, etc.) minus one. Mathematically, compound interest is given as;

C.I = [P (1 + i)n] – 1] – P

Here, C.I is the compound interest

P is the initial principal

I is the interest rate

N is the number of compounding periods.

For clarity, let us assume that John who is a lender has set up a 10% interest on a $10000 loan with a quarterly compounding period for two years. The compound interest in this case would be given as;

[$10,000 (1 + 0.10)8 – 1] = $21,434.

### Compounding Interest Growth

Using John’s imaginary loan, we can see that the interest payout for all 8 payment periods are not the same. If it were to be simple interest, the payout would have been an additional $8000 ( $1000 for each 8 payment periods). However, since compound interest deals with the “interest on interest” rule, one can only expect that the longer the compounding periods, the higher each individual interest payment would get. Compound interest growth rate can be highlighted by the use of tables and charts, as they help to create basic understanding of tough concepts.

Most investment managers and speculators are bonded to the principles of compound interest as it is a better and safer way of building income in the long-run. This is because the “accruement is done on each accruement,” and this leads to a bigger payout over time. For instance; a $300,000 investment with 3% semi-annual interest rate over a period of 20 years would amount to $978,610 compared to a simple interest rate with the same numbers which will amount to $480,000 over the same 20years. Thus, one can see that compound interest is far better than simple interest, however there are some implications and conditions to this interest type which we’ll look at later on.

### Tools for Calculating Compounding Interest: Microsoft Excel

Rather than trying to work as a mathematician while calculating compound interest, one can make use of Microsoft excel which is far easier but removes the fun of calculating financial numbers. You can also make use of handheld calculators and smaller softwares for calculating compounding interest, but if you’re looking at bigger values (especially higher compounding periods and longer years), then you might want to use your Office Subscription for that one.

Here are the basic detailed steps to calculate compound interest using Microsoft Excel or related Excel tools:

- Step one implies multiplying the new balance from each year by the interest rate. This is mostly for an annually compounding interest. Suppose you deposit $500 into a savings account with a 4% interest annual rate, and you’re looking to see how much it’d turn out to be in the next 4 years, then you simply need to create cells in your Excel tool. First, configure cell A1 as your “Year” and Cell B1 as your new yearly balance (Balance). Switch to cell A2 and configure years 0 to 4 from A2 to A6 ( simply just write 0 down to four in each cell starting at A2 and ending at A6). Input your balance for year zero, which in this case is your initial starting deposit of $500. Thus, enter $500 into B2. Next, simply type in this command “=B2*1.04” into B3 without the apostrophe. Do this for B4, i,e, “=B3*1.04”. For B5, follow the exact format. B6 would automatically generate a value which would be your “=B5*1.04). Thus, whatever you get in B7 would be your compound interest, which in this case, it’d be $585. Thus your compound interest value is equal to $585 – $500 = $85.
- Step two is quite simpler than step one as it allows you to just type in a series of numbers and the calculation would start. Here, you’ll need to make use of [P (1+i)n -P], where P is your initial deposit, I is the interest rate and n is total number of compounding periods. Using the same details as the ones in step one, simply type in Principal in Cell A1, interest in Cell A2, and Compound Periods in Cell A3. Now, put 500 in Cell B1, “.04” in Cell B2 (no apostrophe), and “4” in Cell B3 (no apostrophe). To calculate your compound interest, scroll to Cell B4 and type in the command; “=(B1*(1+B2)^B3)-B1.” This will give you $85.
- Step three requires that you generate a macro function to make computing easier. Here, you’re required to power up the Visual Basic Editor first (you can find this in the developer’s tab). Now click on the insert menu and click on Module. After this, type in “Function Compound_Interest(P As Double, i As Double, n As Double) As Double” in the first line. On the second line, hit the tab keyed type in “Compound_Interest = (P*(1+i)^n) – P”. On the third line, type in “End Function.” This creates a macro function to compute interest rate. Now, from this same excel sheet, enter “Compound Interest” in Cell A6 and enter ‘=Compound_Interest(B1,B2,B3)”. This will give you a value of $85, which is the same with those of step one and two.

### Compound Interest Calculation Rate

The rate of accruement in compounding interest varies depending on the set frequency. It can be as low as per day or as high as a year (daily and annually). All financial instruments have their compounding periods to reduce the risk of variety. For savings accounts in a financial bank, the generally used compounding frequency is the daily period. For a CD, the frequency can be daily, monthly or even twice per year (semi-annually). In the case of money markets (or liquid markets) frequency is usually daily although there might be some exceptions. Mortgage payments, equity loans, home loans, personal and business loans, and credit cards all make use of the monthly compounding frequency. There are however different timeframes in which the interest would be added to the existing balance. The crediting of interests in most cases differs from the accruement frequency. In the case of a bank’s savings account, interest is accrued on a daily basis but credited on a monthly basis. Thus, the account won’t be eligible to earn additional interests until the total daily compounded interest for that month has been credited to the principal balance.

In some cases, banks might offer continuously compounding interest, which is different from the normal compound interest. In this case, instead of setting a specific crediting period, the bank will add the compounded interests to the account at any possible time. It is important to note that this accruement isn’t necessary higher than the daily compounding frequency unless you wish to deposit and withdraw money in the same day.

Compound interest is totally beneficial to investment managers and lenders but the same cannot be said for borrowers. This is because compounding makes the loan get bigger. While the lender would certainly be happy with this, the borrower won’t necessary be satisfied as it means that he has to pay a sum way higher than what he borrowed.

### Time Value of Money

Time value of money simply refers to how much time affects money. This is the theory behind the saying that time is typically money. In compound interest, basic and advanced understanding of the time value of money by investors would help them increase their wealth allocation and optimize their income. In understanding this concept, one has to search for the future value (FV) and the present value (PV). Mathematically, we can find both concepts by;

FV = PV (1 + i)n and PV = FV / (1 + i)n

Let us assume that John takes a loan of $5,000 which is compounding at 2.5% semi-annually for a period of 5 years.

The future value would be given as;

= $5,000 (1 + 0.025)10 = $6400

And the present value would be given as;

=$6400 (1 + 0.025)10 = $5000

Here we see that both the future value and present value are related.

### Rule of 72

The Rule of 72 is a very important concept in compound interest. This rule calculates the time required for an investment to double at a designated interest rate “i” and it is mathematically symbolized as “72/i.” The Rule of 72 is only applicable to annual compounding interests.

In real life situations, an investment with 6% annual interest rate would double in 12 years, while one with 8% annual interest rate would double in nine years.

### Compound Annual Growth Rate (CAGR)

Compound annual growth rate is usually utilized on financial instruments that require computation of a single growth rate over a designated duration. For instance, if John’s portfolio has grown from $12,000 to $20,000 in six years, the compound annual growth rate would be denoted via the PV which is -$12,000 and the FV which is $20,000 with the compounding periods which is 6 in this case. Knowing all these values, one can easily compute the CAGR using an Excel calculator.

(PV is generally denoted with a negative sign as seen in our -$12,000 because the amount is actually flowing out and non-retractable till the designated duration is up. FV however doesn’t need to have an opposite sign since this is the amount that John will realize if inflation is taken out of the equation).

### CAGR Application in Real Life

- Compound annual growth rate is used in calculating mutual funds returns, stocks and investments portfolio returns over a designated period. This tool is also utilized in determining if a mutual fund manager or portfolio manager has surpassed the market’s return rate over a given time period. For instance, a market index (say S&P or the Dow Jones) which has acquired returns of 12% over a 10-years, and a manager who has acquired 14% over the same 10-years. In this case, the manager is said to have outperformed the market and would certainly deserve praises.
- CAGR can also be used in computing the potential or projected growth of investment portfolios over a given period of time, which is important for savings purposes in cases of retirements. Let us take a look at three examples given below

### First Example

An investor who prefers low risk is very contented with a 5% annual return on her portfolio. Currently, his portfolio which is valued at $300,000 would grow to $495,989 in 20years. On the other hand, an investor who loves high risk and high returns that is looking for 15% annual return on his portfolio would realize $4,609,961 after the same period of 20years.

### Second Example

For individuals looking to satisfy future desires or wants, the compound annual growth rate can be used to calculate how much money needs to be put off for their objective. For instance, a woman who wished to save $50,000 over 10 years for a down payment on a house would need approximately $4,165 each year if they’re getting a CAGR of 4% on their savings. In a situation where they’re looking to take more risks, put to the point of 5%, they’ll have to save $3,975 per year for a period of 10 years.

### Third Example

Compound annual growth rate is also used in showing the advantages of investing when younger compared to when you grow older in life. For a 20-year old looking to save $1million by the age of 60, he or she would be looking at putting off $6,462 per year in order to get that amount. Someone who is 35 would have to save $18,277 yearly to get that same amount at the age of 60.

- Compound annual growth rate is also used substantially in economic reports and data. For instance; China’s per-capita GDP went up to $6,091in 2012 from $193 in 1980. Here, the CAGR denoted by “i” would be 11.4% over a 32-year period.

### Magic of Compounding

According to Albert Einstein, , the magic of compounding in compound interest is man’s greatest invention and the 8th wonder of the world. While it is true that compounding is good to the lender and investor, the reverse is case for the borrower or someone in debt. An instance would be a credit card loan or debt. A credit card debt of $30,000 with a compounding interest of 15% monthly would result in an additional $4,500 for the first month. This will in turn lead to a debt of $130,507 if left for a whole year.

For the lender and the investor, compounding interest is a great deal as an investment of $30,000 with 15% monthly compounding interest will lead to $130,507. Thus, it is undeniable to say that the magic of compounding can lead to wealth creation and combating inflation. It is also useful in taking care of income-related factors such as the increase in the cost of living, and purchasing power.

The best source or investment for compound interest is a mutual fund as it allows investors to reinvest dividends which will lead to accruement of higher interest payouts per each reinvested dividend. This way, the investor gets to benefit from more interest payments and also grow his or her portfolio in the process.

For emphasis, let us consider a mutual fund portfolio with an initial principal deposit of $5,000 and an annual addition amount of $2,400. If the average annual interest rate is 12% and the mutual fund is expected to last for up to 30 years, the value of the fund after the duration is up would be $798,500. Here, the compound interest would be the different between the actual deposit with the yearly additions and the future value of the investment. So, by contributing $77,000 ($2,400 x 30years), the compound interest would be $721,500 before taxes.

### Similar Investment Tactics

Any portfolio manager or investor who chooses a reinvestment plan in a brokerage account is said to be using the power of compound interest in any security which he invests in. Investors can also experience compounding with purchases of zero-based coupon payments bonds (bonds where the interests are not realized before the expiration of the duration). Other bonds (traditional bonds) do not qualify for compound interests since the interests are typically paid out semi-annually as coupons. Zero-Coupon bonds on the other hand are not qualified for semi-annual coupons, and this is the reason why compounding works great on it till its period of maturity.

In mortgages, compound interest also works when repaying loans. If you split your mortgage payment into two per month instead of paying once, you get to reduce the amortization period and save off a huge sum of interest.

### Simple and Compound Interest in Loans

Differentiating simple and compound interest in loans should be a tough task, it is however important as this allows you to know what you are paying for and how. The Truth in Lending Act (TILA) requires that creditors disclose loan contractual terms and conditions to their potential borrowers, with emphasis based on the total dollar amount which a borrower is required to pay over the lifetime of the loan as well as whether the interest is a simple interest (accrues on the initial loan amount) or a compounding interest (accrues on new balance).

The TILA also requires that such creditors reveal a comparison of a loan’s interest rate to annual percentage rate (APR). The APR converts the financial charges of your loan and adds up all the fees to a simple interest rate. The only difference between interest rates and APR is that APR adds up bogus fees to your simple interest rate, while interest rate in mortgages accrues using compound interest (interest on interest).

### References for “Compound Interest”

https://en.wikipedia.org/wiki/Compound_interest

https://www.investopedia.com/terms/c/compoundinterest.asp

www.moneychimp.com/calculator/compound_interest_calculator.htm

https://www.mathsisfun.com/money/compound-interest.html

https://www.thecalculatorsite.com/articles/finance/compound-interest-formula.php

https://www.investor.gov/additional-resources/free…/compound-interest-calculator

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