Simple vs. Compound Interest: Definition and Formulas
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WATCH: What is Compound Interest?
Compound interest is calculated on the initial principal and on the accumulated interest of previous periods of a deposit or loan. Compound interest can be thought of as “interest on interest,” and will make a deposit or loan grow at a faster rate than simple interest, which is interest calculated only on the principal amount. This video walks through the how to calculate compounding interest and what differentiates it from simple interest.
Reviewed by Margaret JamesFact checked by Suzanne KvilhaugReviewed by Margaret JamesFact checked by Suzanne Kvilhaug
Types of Interest
Interest is defined as the cost of borrowing money. It can also be the rate paid for money on deposit, as in the case of a certificate of deposit. Interest can be calculated in two ways: simple interest or compound interest.
- Simple interest is calculated on the principal, or original, amount of a loan.
- Compound interest is calculated on the principal amount and the accumulated interest of previous periods and can therefore be referred to as “interest on interest.”
There can be a big difference in the amount of interest payable on a loan if interest is calculated on a compound basis rather than on a simple basis. But the magic of compounding can work to your advantage when it comes to your investments. It can be a potent factor in wealth creation.
Simple interest and compound interest are basic financial concepts but becoming thoroughly familiar with them may help you make more informed decisions when you’re taking out a loan or investing. Cumulative interest can also help you choose one bond investment over another.
Key Takeaways
- Interest can refer to the cost of borrowing money in the form of interest charged on a loan or to the rate paid for money on deposit.
- Simple interest is only charged on the original principal amount in the case of a loan.
- Simple interest is calculated by multiplying the loan principal by the interest rate and then by the term of a loan.
- Compound interest multiplies savings or debt at an accelerated rate.
- Compound interest is interest calculated on both the initial principal and all of the previously accumulated interest.
Simple Interest Formula
The formula for calculating simple interest is:
Simple Interest=P×i×nwhere:P=Principali=Interest raten=Term of the loan
The total amount of interest payable by the borrower is calculated as $10,000 x 0.05 x 3 = $1,500 if simple interest is charged at 5% on a $10,000 loan that’s taken out for three years. Interest on this loan is payable at $500 annually or $1,500 over the three-year loan term.
Compound Interest Formula
The formula for calculating the total amount paid on a loan with compound interest is:
A=P(1+nr)ntwhere:A=Final amountP=Initial principal balancer=Interest raten=Number of times interest appliedper time periodt=Number of time periods elapsed
Compound Interest equals the total amount of principal and interest in the future, or future value, less the principal amount at present, referred to as present value (PV). PV is the current worth of a future sum of money or stream of cash flows given a specified rate of return.
What would the amount of interest in the simple interest example be if it was charged on a compound basis?
Interest=$10,000((1+0.05)3−1)=$10,000(1.157625−1)=$1,576.25
The total interest payable over the three-year period of this loan is $1,576.25, unlike simple interest, but the interest amount isn’t the same for all three years because compound interest also considers the accumulated interest of previous periods. Interest payable at the end of each year is shown like this:
| Year | Opening Balance (P) | Interest at 5% (I) | Closing Balance (P+I) |
| 1 | $10,000.00 | $500.00 | $10,500.00 |
| 2 | $10,500.00 | $525.00 | $11,025.00 |
| 3 | $11,025.00 | $551.25 | $11,576.25 |
| Total Interest | $1,576.25 |
Compounding Periods
The number of compounding periods makes a significant difference when calculating compound interest. The higher the number of compounding periods, the greater the amount of compound interest generally is. The amount of interest accrued at 10% annually will be lower than the interest accrued at 5% semiannually for every $100 of a loan over a certain period. This will in turn be lower than the interest accrued at 2.5% quarterly.
The variables “i” and “n” within the parentheses have to be adjusted in the formula for calculating compound interest if the number of compounding periods is more than once a year.
“I” or interest rate has to be divided by “n,” the number of compounding periods per year. “N” has to be multiplied by “t,” the total length of the investment, outside the parentheses. So i = 5% (i.e., 10% ÷ 2) and n = 20 (i.e., 10 x 2) for a 10-year loan at 10% where interest is compounded semiannually: the number of compounding periods = 2.
You would use this equation to calculate the total value with compound interest:
Total Value with Compound Interest=(P(n1+i)nt)−PCompound Interest=P((n1+i)nt−1)where:P=Principali=Interest rate in percentage termsn=Number of compounding periods per yeart=Total number of years for the investment or loan
This table demonstrates the difference the number of compounding periods can make over time for a $10,000 loan taken for a 10-year period.
| Compounding Frequency | No. of Compounding Periods | Values for i/n and nt | Total Interest |
| Annually | 1 | i/n = 10%, nt = 10 | $15,937.42 |
| Semiannually | 2 | i/n = 5%, nt = 20 | $16,532.98 |
| Quarterly | 4 | i/n = 2.5%, nt = 40 | $16,850.64 |
| Monthly | 12 | i/n = 0.833%, nt = 120 | $17,059.68 |
Other Compound Interest Concepts
Compound interest doesn’t only relate to loans.
Time Value of Money
Money isn’t “free” but has a cost in terms of interest payable so it follows that a dollar today is worth more than a dollar in the future. This concept is known as the time value of money and it forms the basis for relatively advanced techniques like discounted cash flow (DFC) analysis.
The opposite of compounding is known as discounting. The discount factor can be thought of as the reciprocal of the interest rate. It’s the factor by which a future value must be multiplied to get the present value. The formulas for obtaining the future value (FV) and present value (PV) are:
FV=PV×[n1+i](n×t)PV=FV÷[n1+i](n×t)where:i=Interest rate in percentage termsn=Number of compounding periods per yeart=Total number of years for the investment or loan
The Rule of 72
The Rule of 72 calculates the approximate time over which an investment will double at a given rate of return or interest “i.” It’s given by (72 ÷ i). It can only be used for annual compounding but it can be very helpful in planning how much money you might expect to have in retirement.
An investment that has a 6% annual rate of return will double in 12 years (72 ÷ 6%). An investment with an 8% annual rate of return will double in nine years (72 ÷ 8%).
Compound Annual Growth Rate (CAGR)
The compound annual growth rate (CAGR) is used for most financial applications that require the calculation of a single growth rate over a period.
What is the CAGR if your investment portfolio has grown from $10,000 to $16,000 over five years? PV = $10,000, FV = $16,000, and nt = 5 so the variable “i” has to be calculated. It can be shown that i = 9.86% using a financial calculator or Excel spreadsheet.
Your initial investment (PV) of $10,000 is shown with a negative sign according to the cash flow convention because it represents an outflow of funds. PV and FV must necessarily have opposite signs to solve “i” in the above equation.
Real-Life Applications
CAGR is used extensively to calculate returns over periods for stocks, mutual funds, and investment portfolios. It’s also used to ascertain whether a mutual fund manager or portfolio manager has exceeded the market’s rate of return over a period. A fund manager has underperformed the market if a market index has provided total returns of 10% over five years but the manager has only generated annual returns of 9% over the same period.
CAGR can also be used to calculate the expected growth rate of investment portfolios over long periods. This is useful for purposes such as saving for retirement. Consider these examples:
- A risk-averse investor is happy with a modest 3% annual rate of return on their portfolio. Their present $100,000 portfolio would therefore grow to $180,611 after 20 years. But a risk-tolerant investor who expects an annual rate of return of 6% on their portfolio would see $100,000 grow to $320,714 after 20 years.
- CAGR can be used to estimate how much must be stowed away to save for a specific objective. A couple who would like to save $50,000 toward a down payment on a condo over 10 years would have to save $4,165 per year if they assume an annual return (CAGR) of 4% on their savings. They would have to save $3,975 annually if they’re prepared to take on additional risk and expect a CAGR of 5%.
- CAGR can also be used to demonstrate the virtues of investing earlier rather than later in life. A 25-year-old would have to save $6,462 per year to attain their goal if the objective is to save $1 million by retirement at age 65 based on a CAGR of 6%. A 40-year-old would have to save $18,227 or almost three times that amount to attain the same goal.
Additional Interest Considerations
Make sure you know the exact annual percentage rate (APR) on your loan because the method of calculation and number of compounding periods can have an impact on your monthly payments.
Important
Banks and financial institutions have standardized methods to calculate interest payable on mortgages and other loans but the calculations may differ slightly from one country to the next.
Compounding can work in your favor when it comes to your investments but it can also work for you when you’re making loan repayments. Making half your mortgage payment twice a month rather than the full payment once a month will end up cutting down your amortization period and saving you a substantial amount of interest.
Compounding can work against you, however, if you carry loans with very high rates of interest like credit card or department store debt. A credit card balance of $25,000 carrying at an interest rate of 20% compounded monthly would result in a total interest charge of $5,485 over one year or $457 per month.
Which Is Better, Simple or Compound Interest?
It depends on whether you’re investing or borrowing. Compound interest causes the principal to grow exponentially because interest is calculated on the accumulated interest over time as well as on your original principal. It will make your money grow faster in the case of invested assets. Compound interest can create a snowball effect on a loan, however, and exponentially increase your debt. You’ll pay less over time with simple interest if you have a loan.
What Are Some Financial Products That Use Simple Interest?
Some personal loans and simpler consumer products use simple interest. Most bank deposit accounts, mortgages, credit cards, and some lines of credit tend to use compound interest.
How Often Does Interest Compound?
Interest compounding periods can be daily, monthly, quarterly, or annually. The higher the number, the greater the effect of compounding.
Is Compound Interest Considered Income?
Yes. Compound interest on some types of investments such as savings accounts or bonds is considered income.
The Bottom Line
Get the magic of compounding working for you by investing regularly and increasing the frequency of your loan repayments. Familiarizing yourself with the basic concepts of simple interest and compound interest will help you make better financial decisions, saving you thousands of dollars and boosting your net worth over time.
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