Interest / Compound interest / Interest formula / Interest calculator
Encyclopedia of Business Terms and Methods, ISBN 978-1-929500-10-9. Copyright © 2011 by Marty J.Schmidt. Revised 3 March 2012.
The Meaning of Interest, Compounding, and Related Terms
Having the use of money for a period of time has value—a value that is real and measurable. With loans or investments, borrowers are typically charged and lenders (or investors) typically receive payment for the use of funds. The primary component of these payments is defined as interest. Interest payments are calculated as a function of at least two factors:
1. A percentage of the amount borrowed or invested, covering a specific time period. The percentage figure is the interest rate and the amount borrowed or invested is the principal.
2. The number of interest-paying periods covering the duration of the loan or investment.
This entry presents the basic terms and concepts involved in calculating interest payments and interest earnings in the context of interest-related terms including:
• Interest | • Compounding | • Nominal interest rate |
Interest earned and interest paid are central tools in the practice of modern finance—making the best use of the organization's financial assets. Financial officers try to put borrowed funds to work so as to return more than the costs of borrowing. Individual Investors and business people of all kinds attempt to do the same. The challenge for all concerned is that interest rates—and therefore interest earnings and interest costs—are often subject to change over time, depending on factors that are not entirely predictable. The financial mathematics in the sections below are relatively simple and clear, but the predicted financial results are certain only when the interest rates involved are known and certain.
• Formulas: Calculating One Period Interest and Compound Interest for Multiple Periods
• The Effect of Compounding Frequency and Continuous Compounding on Compound
Interest Growth
• Interest Rates for Comparing Loan Costs and Investment Returns
– Nominal interest Rate (or Annual Percentage Rate, APR)
– Effective Interest Rate (or, Annual Effective Rate, AER)
– Real Interest Rates
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Formulas: Calculating One Period Interest and Compound Interest for Multiple Periods
- How much will investors (or depositors) earn, in interest, for the use of their funds? How is future value calculated? What is the formula for future value?
In the case of a loan or an investment (such as an interest-paying bank deposit), interest calculations begin with a stated interest rate and the time period it covers, e.g., "8.0% per year." The calculation also needs the initial amount invested (the principal), and the number of such periods that the funds will stay on deposit (e.g., 1 year, or 10 ten years).
The formula at left uses these data to answer the question above with a Future Value (FV). For an investment, FV is the sum of the principal returned to the investor when the last investment period is over, plus all interested earned. (Note that formulas here intentionally use the same symbols used elsewhere in this encyclopedia tor other time-value-of-money concepts such as discounted cash flow).
Interest calculation example, one year: Consider a one-year $100 investment, at an annual interest rate of 5.0%. What is the future value (FV) of this investment after one year? How much interest is earned? For this example:
PV = $100.00
i = 5.0% = 0.05 per year
n = 1 year
Future Value (FV)
FV = PV (1 + i )n
= $100.00 ( 1.0 + 0.05 )1
= $100.00 ( 1.05)
= $105.00
Total interest earned = FV – PV
= $105.00 – $100.00
= $5.00
When earned interest is left on deposit after one period, it is added to the principle for the next period's interest calculation. That is, across multiple periods, earned interest earns interest on itself.
- Earning interest on previously earned interest is called compounding.
- The resulting interest total is compound interest.
- Each interest-earning period in which compounding occurs is a compounding period.
Example future value calculation with compounding: What is the FV after two years of a $100 investment, paying 5% per year interest? What is the FV after 10 years, at the same interest rate?
For the two-year case:
PV = $100.00
i = 5.0% = 0.05 per year
n = 2 years
Future Value after two years (FV2)
FV2 = PV (1 + i )2
= $100.00 ( 1.0 + 0.05 )2
= $100.00 ( 1.1025)
= $110.25
For the ten-year case, n =10 years
FV10 = PV (1 + i )10
= $100.00 ( 1.0 + 0.05 )10
= $100.00 ( 1.6289)
= $162.89
Exhibit 1 below shows that when future value are plotted as a function of the number of periods, upward curving lines result. This is the nature of exponential functions, such as the FV formulas above. The exhibit also shows that a few percentage points difference in interest rates (5% vs 8%) leads over many compounding periods to very large differences in future value.
Exhibit 1. Future value growth for a $100 principal, compounded over 20 periods, using two different interest rates. A few percentage points difference in interest rates (5% vs. 8%) leads to an increasingly large difference in future values across many compounding periods. |
For those who wonder why the exponent for (1+ i ) works this way for compound interest, note that the formula FV2 = PV (1 + i )2 is mathematically equivalent to taking FV1 (the FV after one period, including the first year's interest), and making that the new PV for another interest calculation for the second year. I.e.,
PV (1 + i )2 = PV ( 1+ i )1 (1 + i )1
The expression on the right is transformed into the expression on the left by recognizing that two identical ( 1 + i ) terms are multiplied by stating the term just once and adding their exponents ( 1 + 1 = 2 ). To obtain FVs based on compounded interest, when the interest rate i stays the same from period to period, simply set the exponent for a single ( 1 + i ) term equal to the number of periods, as shown in the example
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The Effect of Compounding Frequency and Continuous Compounding on Compound Interest Growth
- What is future value of a two-year investment when the compounding periods are one month instead of one year? What is FV if compounding is performed daily? Continuously?
Interest on financial investments is often calculated, or compounded, on a semiannual, quarterly, monthly, or daily basis, as well as on an annual basis. Compounding may even occur on a "continuous" basis The examples above used annual (1-year) compounding periods, but shorter periods mean that compounding occurs with a higher frequency.
For mufti-period loans or investments, compounding frequency also has an impact on final future value. A ten-year $100 investment paying 5.0% for each annual period, leads to the FV of $162.89 after 10 years, as shown in the example above. A ten year $100 investment paying interest compounded monthly, at a monthly rate one-twelfth the annual rate (0.4167%.per month), leads to a FV of $164.70 after 120 months (10 years).
When loans or investments with different compounding frequencies are compared to each other, however, an annual rate of some kind for each is brought into view to facilitate comparison. Several approaches for quoting this annual rate are defined and illustrated in the following section. The examples in this section, however, refer to an annual rate usually called simply the nominal interest rate:
Nominal interest rate = ( Interest rate per period ) ( Number of periods per year )
When interest is calculated with monthly compounding periods at, say, 1.0% per period, the nominal interest rate is 12.0%. That is, 12 x 1.0% = 12.0%. Using the nominal interest rate concept, the formula for future value with compound interest can be modified to recognize different compounding frequencies:
Note that In the first FV formula, the expression i/q is the interest rate per period. The exponent Yq is the total number of compounding periods. The second FV formula calculates the result under continuous compounding—compound interest as though an infinite number of infinitesimally small compounding periods are used.
Example: Future Value With Different Compounding Frequencies.
Consider a 10-year interest-paying bank deposit of $1,000, where the nominal (annual) interest rate is 8.0%, and where interest is compounded monthly. Using the symbols above for this case:
PV = $1,000.00
i = 8.0% nominal interest rate per year
q = 12 compounding periods per year
Y = 10 years of compounding
FV = PV ( 1 + i / q )Yq
= $1,000 ( 1.0 + 0.08 / 12 )(10)(12)
= $1,000 (1.00667)120
= $1,000 (2.21964)
= $2,219.64
For the same deposit with continuous compounding:
PV = $1,000.00
i = 8.0% nominal interest rate per year
Y = 10 years of compounding
e = 2.71828182845904... (e is a constant and always has this value)
FV = PV eYi
= ($1,000) e(10)(0.08)
= ($1,000) e0.8
= $2,225.54
Note that expressions involving the natural logarithm constant e can be implemented easily in Microsoft Excel with the EXP() function. The Excel version of the last FV formula is:
= 1000*EXP(10*0.08)
Table 1 below compares FV results for the example $1,000, ten-year deposit using a nominal 8.0% interest rate with different compounding frequencies:
| Compounding Frequency | Periods per Year | Nominal Interest | Interest per Period | Years | Future Value |
|---|---|---|---|---|---|
| Annual | 1 | 8.0% | 8.0% | 10 | $2,158.92 |
| Semi Annual | 2 | 8.0% | 4.0% | 10 | $2,191.12 |
| Quarterly | 4 | 8.0% | 2.0% | 10 | $2,208.04 |
| Monthly | 12 | 8.0% | 0.666667% | 10 | $2,219.64 |
| Daily | 365 | 8.0% | 0.021918% | 10 | $2,225.35 |
| Continuous | ∞ | 8.0% | ≈ 0.0% | 10 | $2,225.54 |
| Table 1. Future value increases as frequency increases, with continuous compounding leading to the maximum FV for a given nominal interest rate. | |||||
From the table it is clear that FV increases as compounding frequency increases, approaching it's maximum limit when the frequency is infinite (continuous compounding). However, the important aspects of the relationship between compounding frequency and future value are easier to grasp when these data are shown in graphical form as shown in Exhibit 2, below:
Exhibit 2. Future value as a function of compounding frequency. Clearly, future value comes very close to its maximum value when compounding frequency goes from monthly compounding (frequency = 12) to daily compounding (frequency=365). The added value of continuous compounding over daily compounding is negligible. |
Working examples of the computations in this entry are illustrated and explained further in the spreadsheet tool, Financial Metrics Pro, along with suggestions for spreadsheet implementation.
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Interest Rates for Comparing Loan Costs and Investment Returns
Examples above refer to a nominal interest rate. That is one basis for comparing interest costs and interest returns from different compounding plans. There are other interest rate methods in common usage, however, meant for approximately the same purpose.
Unfortunately, some of these rate methods have several names, and one of the methods (annual percentage rate, or APR) not only has several names, but is defined differently in different countries. As a result, the distinctions among different rate terms can seem obscure or confusing. To avoid confusion in the following discussion, keep in mind that all these interest-related terms fall essentially into just three classes, characterized by the terms "Nominal," "Effective," and "Real."
Nominal Interest Rate
Nominal Annual Rate
Nominal Annual Percentage Rate
Nominal APR
All four terms in this section heading may be considered equivalent. The nominal interest rate (or nominal annual rate, nominal annual percentage rate, or nominal APR) . . .
- Is a rate (percentage) for describing loan or investment interest on an annual basis.
- Is not adjusted to reflect inflation, not adjusted to reflect the contributions of compounding, and is not adjusted to reflect other investing or borrowing costs.
- Is a term used primarily when the compounding frequency (e.g., 12 times per year, or monthly) does not equal the unit of time used for describing the rate (e.g., "8% per year").
- Is simply the number of compounding periods per year multiplied by the interest rate per period. How is nominal interest rate calculated? What is the formula for nominal interest rate?
Nominal interest rate = ( Interest rate per period ) ( Number of periods per year )
Example: When interest is calculated with monthly compounding periods at 1.0% per period, the nominal interest rate is 12.0%. That is, 12 x 1.0% = 12.0%
Nominal interest rates provide a quick and easily understood means for comparing loans or investments with different compounding frequencies. They are called "nominal" because they are not adjusted to reflect inflation, compounding, or other costs (as are the rates below).
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Effective Interest Rate
Annual Equivalent Rate, AER
Annual Effective Rate, AER
Annual Effective Yield
Effective Annual Yield
Annual Percentage Rate, APR
All terms in this section heading may be considered essentially equivalent, except for some slight differences in usage, explained below.
- Would an investor prefer to receive interest compounded daily or compounded annually? How is effective interest rate calculated?
A ten-year deposit of $1,000 at a nominal rate of 8%, leads to a future value of $2,159 when interest is compounded annually. The same deposit at an 8% nominal rate leads to a future value of $2,225 with daily compounding. .
Both scenarios use the same principal, same deposit time, and same nominal rate, but clearly the daily compounding effectively returns more. Interest rate methods named with the word "effective" attempt to recognize the contribution of compounding frequency (and, sometimes, other factors as well) so that comparisons of different investing or borrowing plans can be compared more appropriately.
The effective interest rate concept takes into account the impacts of more frequent compounding. In its simplest form, the effective interest rate (r) for a loan or investment is given by the formulas below:
These formulas require a nominal interest rate ( i ) and the number of compounding periods per year (here, n = compounding periods per year) as input.
Effective interest rate calculation examples: For a $1000 deposit at a nominal interest rate of 8.0% ( i = 8.0%), compounded annually ( n -1 ), the effective interest rate is the same as the nominal rate:
r = ( 1 + i / n ) n – 1
= (1 + 0.08)1 – 1
= 0.08 = 8.0%
However, the same investment at the same 8% nominal interest rate, leads to a different effective rate with daily compounding ( n =365):
r = ( 1 + i / n ) n – 1
= (1 + 0.08 / 365)365 – 1
= ( 1.00021917808)365 – 1
= 1.08327757178281 – 1 = 0.0832775718
= about 8.3278%
What these results say is that an investment of $1,000 with a nominal interest rate of 8.33% compounded annually would have a future value equal to a $1,000 investment with a nominal rate of 8.00% compounded daily. With daily compounding, in other words, the effective annual rate is 8.3277%%.
The maximum effective interest rate occurs under continuous compounding. For a continuously compounded investment, with the same 8% nominal interest rate, the effective rate is:
r = ei – 1
= (2.7182828…)0.08 – 1
= 1.083287068... – 1.0 = 0.083287068
= 8.3287068%
= about 8.3287%
Clearly, the differences in effective interest rates between the daily compounding situation ( n = 365) and continuous compounding are small—about one one-thousandth of one percent.
Table 2, below parallels Table 1 above, showing different compounding frequencies for a deposit with an 8.0% nominal interest rate, but now showing how effective interest rate changes with compounding frequency:
| Compounding Frequency | Periods per Year | Nominal Interest | Interest per Period | Effective Int. Rate | 10- Year FV |
|---|---|---|---|---|---|
| Annual | 1 | 8.0% | 8.0% | 8.0000000% | $2,158.92 |
| Semi Annual | 2 | 8.0% | 4.0% | 8.1600000% | $2,191.12 |
| Quarterly | 4 | 8.0% | 2.0% | 8.2432160% | $2,208.04 |
| Monthly | 12 | 8.0% | 0.666667% | 8.2999507% | $2,219.64 |
| Daily | 365 | 8.0% | 0.021918% | 8.3277572% | $2,225.35 |
| Continuous | ∞ | 8.0% | ≈ 0.0% | 8.3287068% | $2,225.54 |
| Table 2. Future values and effective interest rates, as compounding frequency increases. The increases, for a deposit of $1,000 paying a nominal interest rate of 8.0%example deposit (investment) is made with an 8.0% nominal rate. | |||||
Effective interest as illustrated here is called, variously: Effective interest rate, Annual effective rate (AER), and Annual equivalent rate (also AER). When the rate is calculated for an interest paying investment, such as a bank certificate of deposit (CD), the terms Annual effective yield or Effective annual yield may be used for the same calculation. (This usage of "yield" should not be confused with the different calculation used in evaluating bond investments, also called Yield or Yield to maturity. For more on these latter terms in the context of bond investments, see the encyclopedia entry bond.)
The effective Interest rate is also at the center of the interest rate term Annual percentage rate, or APR. Annual percentage rate—as the term is used in most countries—often has a legal definition from the government, meant to provide depositors or investors with an accurate measure for comparing expected returns from different potential investments. As usually defined, APR includes the effective interest rate, but also reflects some borrowing costs not related to interest, such as loan origination fees, periodic maintenance fees, and others. (See the entry cost of borrowing for more on the role of borrowing costs besides interest).
In the U.S., the calculation and disclosure of APR is governed by the Truth in Lending Act (TILA) of 1968 (a federal law). In the US, APR as specified by the TILA is the nominal interest rate, adjust to include specific non-interest costs. The TILA does not attempt to regulate rates and costs that may be charged, but rather, addresses disclosure. For mortgages, for instance, The TILA prescribes that lenders must disclose the APR to applicants within 3 days of applying.
APR is defined in U.K. by the Consumer Credit Act of 1974. This law requires that APR be published and prominently visible for all regulated loans.
The European Union has issued a series of directives over time (e.g., 87/102/CEE and 98/7/EC) requiring member states to move closer to uniformity on the components and calculation of APR, while still allowing some differences between countries. These directives are also intended primarily to regulate disclosure.
In Canada APR (also called Effective interest rate, EIR) must be disclosed for loans, mortgages, and credit card debt, with a calculation that factors in borrowing costs including loan origination fees, account maintenance fees, others.
The Australian Consumer Credit Code (ACCC) of 2003 requires that APR for all consumer loans and credit be stated with a formula that recognizes interest as well as fees (upfront fees, ongoing fees, and exit fees). In Australia this rate is usually called the AAPR (Average annual percentage rate).
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Real Interest Rate
The real interest rate concept is an attempt to adjust stated interest rates for loans or investments to compensate for the effect of inflation. Real interest rate is usually defined simply as the nominal interest rate for a loan or investment minus the inflation rate:
Real interest rate = Nominal Interest Rate – Inflation rate
When inflation is running at an annual rate of, say, 2.0% for an economy, a loan or investment with a nominal interest rate of 8.0% can be considered to have a real interest rate of about 6.0%. When using real interest rates, however, remember (1) that inflation rates can and usually do change somewhat from year to year, and (2) inflation rates differ substantially from country to country.
When the inflation rate is high, interest paid and interest charged tend to tend to be higher and when the inflation rate is low, interest rates are also low. The result is that real interest rates exhibit more stability in changing economies than do the nominal interest rates.
For evaluating historical loans investments, inflation rates can be determined with high confidence. Looking forward in time, however, to evaluate potential real interest rates from investments, the future inflation rates are necessarily estimates that come with uncertainty.
For estimates of inflation rates by country, historical and current, see the country specific economic data, which can be accessed through links on the Solution Matrix Ltd. Economic Data web page. For more on the use of interest rates as economic indicators, or leading indicators, see the encyclopedia entry economic indicator.
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