Simple vs. Compound Interest: Definitions and Formulas (2023)

Interest is defined as the cost of borrowing money, as in the case of interest on a loan balance. Conversely, no interest may be paid on money on deposits, as with certificates of deposit. Interest can be calculated in two ways:simple interesttherenters interest.

  • Simple interestis intended for itheadmaster, or the original loan amount.
  • Renters interestisis calculatedon principal and accumulated interest from previous periods, and can therefore be considered "interest of interest".

There can be a big difference in how much interest must be paid on a loan if the interest is calculated on a compound basis rather than a simple basis. On the bright side, the magic of compound interest can work in your favor when it comes to your investments and can be a powerful factor in wealth creation.

Menssimple interest and compound interestare basic financial concepts, being fully familiar with them can help you make more informed decisions when taking out a loan or investing.

Simple interest formula

The formula for calculating simple interest is:

Simple interest=Pi×I×nwhere:Pi=headmasterI=Interest raten=Term of the loan\begin{aligned}&\text{Simple Interest} = P \time i \times n \\&\textbf{where:}\\&P = \text{Principal} \\&i = \text{Rente} \\ &n = \text{Duration of the loan} \\\end{aligned}Simple interest=Pi×I×nwhere:Pi=headmasterI=Interest raten=Term of the loan

So if simple interest is charged at 5% on a $10,000 loan taken out over three years, then the total amount of interest paid by the borrower is calculated as $10,000 x 0.05 x 3 = $1,500.

Interest on this loan is paid at $500 per year or $1,500 over the three-year term of the loan.

Compound interest formula

The formula for calculating compound interest in one year is:

Renters interest=(Pi(1+I)n)PiRenters interest=Pi((1+I)n1)where:Pi=headmasterI=Interest terms in percentn=Number of compounding periods in a year\begin{aligned} &\text{Compound interest} = \big ( P(1 + i) ^ n \big ) - P \\ &\text{Compound interest} = P \big ( (1 + i) ^ n - 1 \big ) \\ &\textbf{where:}\\ & P= \text{Principal}\\ &i = \text{Interest in percent} \\ &n = \text{Number of compound periods in a year } \ \ \end{match}Renters interest=(Pi(1+I)n)PiRenters interest=Pi((1+I)n1)where:Pi=headmasterI=Interest terms in percentn=Number of compounding periods in a year

Compound interest = the total amount of principal and interest in the future (orfuture value) minus the current capital, which is calledpresent value(PV). PV is the present value of a future sum or flow of moneythe money flowsgiven a specifiedrate of return.

Continuing with the simple interest example, what would the interest amount be if charged on a compounding basis? In this case it would be:

Interesting=$10,000((1+0,05)31)=$10,000(1,1576251)=$1,576,25\begin{aligned} \text{Interest} &= \$10.000 \big( (1 + 0,05) ^ 3 - 1 \big ) \\ &= \$10.000 \big ( 1.157625 - 1 \big ) \\ &= $1.576. \\ \end{στοιχισμένος}Interesting=10 USD,000((1+0,05)31)=10 USD,000(1,1576251)=$1,576,25

While the total interest paid over the three-year term of this loan is $1,576.25, unlike simple interest, the interest amount is not the same for all three years because compound interest also takes into account accumulated interest from previous periods. The interest rates at the end of each year appear in the table below.

YearOpening balance (P)Interest 5% (I)Closing balance (P+I)
110.000,00 $500,00 $10.500,00 $
210.500,00 $525,00 $11.025,00 $
311.025,00 $551,25 $11.576,25 $
Total interest1.576,25 $


CAUTION: What is compound interest?

Complex periods

When calculating compounding, the number of compounding periods makes a significant difference. In general, the higher the number of interest adjustment periods, the higher the interest rate. Thus, for every $ 100 of a loan for a certain period, the amount of interestaccrualat 10% per annum will be lower than accrued interest of 5% semi-annually, which in turn will be lower than accrued interest of 2.5% quarterly.

In the formula for calculating compound interest, the variables "i" and "n" must be adjusted if the number of interest bonus periods is more than once a year.

That is, within the parenthesis, "i" or the interest rate must be divided by "n", the number of compounding periods per year. Outside the parentheses, "n" must be multiplied by "t", the total length of the liner.

For a 10-year loan at 10%, where the interest is compounded semi-annually (number of fixed interest periods = 2), i = 5% (i.e. 10% ÷ 2) and n = 20 (i.e. 10 x 2) ).

To calculate the total value with compound interest, use this equation:

Total value with compound interest=(Pi(1+In)nt)PiRenters interest=Pi((1+In)nt1)where:Pi=headmasterI=Interest terms in percentn=Number of compounding periods per yeart=Total number of years for the investment loan\begin{aligned} &\text{Total value with compound interest} = \Big( P \big ( \frac {1 + i}{n} \big ) ^ {nt} \Big ) - P \\ &\text {Compound interest} = P \Big ( \big ( \frac {1 + i}{n} \big ) ^ {nt} - 1 \Big ) \\ &\textbf{where:} \\ &P = \text{ Principal} \\ &i = \text{Interest in percent} \\ &n = \text{Number of compound periods per years} \\ &t = \text{Total number of years for the investment or loan} \\ \end { adjusted}Total value with compound interest=(Pi(n1+I)nt)PiRenters interest=Pi((n1+I)nt1)where:Pi=headmasterI=Interest terms in percentn=Number of compounding periods per yeart=Total number of years for the investment loan

The following table shows the difference the number of upgrade periods can make over time for a $10,000 loan taken out over a 10-year period.

Composition frequencyNumber of combination periodsValues ​​for i/n and ntTotal interest
Yearly1i/n = 10 %, nt = 1015.937,42 $
Semi-annually2i/n = 5 %, nt = 2016.532,98 $
Quarterly4i/n = 2,5 %, nt = 4016.850,64 $
Monthly12i/n = 0,833 %, nt = 12017.059,68 $

Other terms with compound interest

Time value of money

Since money is not "free" but has a cost in the form of interest to be paid, it follows that a dollar today is worth more than a dollar in the future. This concept is known astime value of moneyand forms the basis for relatively advanced techniques such asdiscounted cash flow(DFC) analysis. The opposite of composition is known asdiscounting. The discount factor can be thought of as the inverse of the interest rate and is the factor by which a future value must be multiplied to obtain the present value.

The formulas for obtaining future value (FV) and present value (PV) are as follows:

FV=PiV×[1+In](n×t)PV=eatV÷[1+In](n×t)where:I=Interest terms in percentn=Number of compounding periods per yeart=Total number of years for the investment loan\begin{aligned}&\text{FV}=PV\times\left[\frac{1+i}{n}\right]^{(n\times t)}\\&\text{PV}=FV \div\left[\frac{1+i}{n}\right]^{(n\times t)}\\&\textbf{where:}\\&i=\text{Interest in percent}\ \ &n =\text{Number of compound periods per years}\\&t=\text{Total number of years for the investment or loan}\end{aligned}FV=PiV×[n1+I](n×t)PV=eatV÷[n1+I](n×t)where:I=Interest terms in percentn=Number of compounding periods per yeart=Total number of years for the investment loan

The rule of 72

TheRule 72calculates the approximate time it will take an investment to double at a given rate of return or interest "i" and is given by (72 ÷ i).It can only be used for annual compounding, but it can be very useful in planning how much money you should expect to have in retirement.

For example, an investment that has an annual return of 6% will double in 12 years (72 ÷ 6%).

An investment with an annual return of 8% will double in nine years (72 ÷ 8%).

Compound Annual Growth Rate (CAGR)

Thecompound annual growth rate(CAGR) is used for most financial applications that require the calculation of a uniform growth rate over a period of time.

For example, if your investment portfolio has grown from $10,000 to $16,000 over five years, what is the CAGR? Essentially, this means that PV = $10,000, FV = $16,000, and nt = 5, so the variable "i" needs to be calculated. Using a financial calculator orExcel regneark, it can be shown that i = 9.86%.

Note that under the cash flow convention, your original investment (PV) of $10,000 is shown with a negative sign as it represents an outflow of funds. PV and FV must necessarily have opposite signs to solve for 'i' in the above equation.

Real applications

CAGR is widely usedto calculate returns over periodsfor shares, mutual funds and investment portfolios. CAGR is also used to determine whether a mutual fund manager or portfolio manager has outperformed the market over a period of time. For example, if a market index has produced a total return of 10% over five years, but a fund manager has generated an annual return of only 9% over the same period, then the manager haswas underperformingthe market.

CAGR can also be used to calculate the expected growth rate of investment portfolios over long periods of time, which is useful for purposes such as retirement savings. Consider the following examples:

  1. A risk-averse investor is happy with a modest 3% annual return on his portfolio. Therefore, their current portfolio of $100,000 will grow to $180,611 after 20 years. Conversely, a risk-tolerant investor who expects a 6% annual return on his portfolio will see $100,000 grow to $320,714 after 20 years.
  2. CAGR can be used to calculate how much needs to be saved to save for a specific goal. A couple looking to save $50,000 over 10 years for a down payment on a condo would need to save $4,165 per year assuming a 4% annual rate of return (CAGR) on their savings. If they are prepared to take on additional risks and expect a CAGR of 5%, they should save $3,975 per year.
  3. CAGR can also be used to show the benefits of investing earlier rather than later in life. If the goal is to save $1 million by retirement at age 65, based on a 6% CAGR, a 25-year-old would need to save $6,462 per year to reach that goal. A 40-year-old, on the other hand, would need to save $18,227, or nearly triple, to reach the same goal.

Additional considerations of interest

Make sure you know exactlyannual rate(APR) for your loan, as the calculation method and the number of upgrade periods can affect your monthly payments. While banks and financial institutions have standardized methods for calculating interest on mortgages and other loans, the calculations may vary slightly from country to country.

Upgrading can work to your advantage when it comes to your investments, but it can also work for you when paying off loans. For example, if you pay half of your mortgage twice a month, instead of making the full payment once a month, you end up shortening your repayment period and saving a significant amount of interest.

Compounding can work against you if you have loans with very high interest rates, such as credit card or department store debt. For example, a $25,000 credit card balance carried at 20% – compounded monthly – would result in a total interest charge of $5,485 for a year, or $457 per month.

Bottom line

Get the magic of it 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, save you thousands of dollars, and increase your net worth over time.

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  1. US Securities and Exchange Commission. "Create settings."

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