What nominal rate compounded semi annually is equivalent to 25 compounded monthly

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Nominal and Effective Interest Rates

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An interest rate takes two forms: nominal interest rate and effective interest rate. The nominal interest rate does not take into account the compounding period. The effective interest rate does take the compounding period into account and thus is a more accurate measure of interest charges.

A statement that the "interest rate is 10%" means that interest is 10% per year, compounded annually. In this case, the nominal annual interest rate is 10%, and the effective annual interest rate is also 10%. However, if compounding is more frequent than once per year, then the effective interest rate will be greater than 10%. The more often compounding occurs, the higher the effective interest rate.

The relationship between nominal annual and effective annual interest rates is:

ia = [ 1 + (r / m) ] m - 1

where "ia" is the effective annual interest rate, "r" is the nominal annual interest rate, and "m" is the number of compounding periods per year.

Example: A credit card company charges 21% interest per year, compounded monthly. What effective annual interest rate does the company charge?

r = 0.21 per year

m = 12 months per year

ia = [ 1 + (.21 / 12) ] 12 - 1

= [1 + 0.0175 ] 12 - 1

= (1.0175)12 - 1 = 1.2314 - 1

= 0.2314 = 23.14%

It may be desired to find the effective interest rate for a period other than annual. In this case, adjust the period for "r" and "m" as needed. For example, if the effective interest rate per semi annual period (every 6 months) is desired, then

r = nominal interest rate per 6 months

m = number of compounding periods per 6 months

and the effective interest rate, isa, per semi-annual period, is:

isa = [ 1 + (r / m) ] m - 1

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Nominal and Effective Interest Rates

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Question 1.

If a lender charges 12% interest, compounded quarterly, what effective annual interest rate is the lender charging?

Choose an answer by clicking on one of the letters below, or click on "Review topic" if needed.

A ia = [ 1 + (0.12 / 12) ] 12 - 1 = (1.01)12 - 1 = 1.1268 - 1 = .1268 = 12.68%

B ia = [ 1 + 0.12 ] 12 - 1 = (1.12)12 - 1 = 3.8960 - 1 = 2.8960 = 289.6%

C ia = [ 1 + (0.12 / 12) ] 4 - 1 = (1.01)4 - 1 = 1.0406 - 1 = .0406 = 4.06%

D ia = [ 1 + (0.12 / 4) ] 4 - 1 = (1.03)4 - 1 = 1.1255 - 1 = .1255 = 12.55%

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Question 2.

If a lender charges 12% interest, compounded monthly, what is the effective interest rate per quarter?

Hint: m = number of compounding periods per quarter

Let i = effective interest rate per quarter.

Choose an answer by clicking on one of the letters below, or click on "Review topic" if needed.

A i = [ 1 + (0.12 / 3) ] 3 - 1 = (1.04)3 - 1 = 0.1249 = 12.49%

B i = [ 1 + 0.03 ] 12 - 1 = (1.03)12 - 1 = 0.4258 = 42.58%

C i = [ 1 + (0.03 / 3) ] 3 - 1 = (1.01)3 - 1 = 0.0303 = 3.03%

D i = [ 1 + (0.03 / 12) ] 3 - 1 = (1.0025)3 - 1 = 0.0075 = 0.75%

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What Is an Effective Annual Interest Rate?

An effective annual interest rate is the real return on a savings account or any interest-paying investment when the effects of compounding over time are taken into account. It also reflects the real percentage rate owed in interest on a loan, a credit card, or any other debt.

It is also called the effective interest rate, the effective rate, or the annual equivalent rate (AER).

Key Takeaways

• The effective annual interest rate is the true interest rate on an investment or loan because it takes into account the effects of compounding.
• The more frequent the compounding periods, the higher the rate.
• A savings account or a loan may be advertised with both a nominal interest rate and an effective annual interest rate.
• The effective annual interest rate is the rate that should be compared between loans and investment rates of return.

The Effective Annual Interest Rate

Understanding the Effective Annual Interest Rate 

The effective annual interest rate describes the true interest rate associated with an investment or loan. The most important feature of the effective annual interest rate is that it takes into account the fact that more frequent compounding periods will lead to a higher effective interest rate.

Suppose, for instance, you have two loans, and each has a stated interest rate of 10%, in which one compounds annually and the other compounds twice per year. Even though they both have a stated interest rate of 10%, the effective annual interest rate of the loan that compounds twice per year will be higher. 

The effective annual interest rate is important because, without it, borrowers might underestimate the true cost of a loan. And investors need it to project the actual expected return on an investment, such as a corporate bond.

Effective Annual Interest Rate Formula 

The following formula is used to calculate the effective annual interest rate:

E f f e c t i v e   A n n u a l   I n t e r e s t   R a t e = ( 1 + i n ) n − 1 where: i = Nominal interest rate n = Number of periods \begin{aligned} &Effective\ Annual\ Interest\ Rate=\left ( 1+\frac{i}{n} \right )^n-1\\ &\textbf{where:}\\ &i=\text{Nominal interest rate}\\ &n=\text{Number of periods}\\ \end{aligned} ​Effective Annual Interest Rate=(1+ni​)n−1where:i=Nominal interest raten=Number of periods​

What the Effective Annual Interest Rate Tells You

A certificate of deposit (CD), a savings account, or a loan offer may be advertised with its nominal interest rate as well as its effective annual interest rate. The nominal interest rate does not reflect the effects of compounding interest or even the fees that come with these financial products. The effective annual interest rate is the real return.

That’s why the effective annual interest rate is an important financial concept to understand. You can compare various offers accurately only if you know the effective annual interest rate of each one.

Example of Effective Annual Interest Rate

Consider these two offers: Investment A pays 10% interest, compounded monthly. Investment B pays 10.1%, compounded semiannually. Which is the better offer?

In both cases, the advertised interest rate is the nominal interest rate. The effective annual interest rate is calculated by adjusting the nominal interest rate for the number of compounding periods that the financial product will undergo in a period of time. In this case, that period is one year. The formula and calculations are as follows:

• Effective annual interest rate = (1 + (nominal rate ÷ number of compounding periods)) ^ (number of compounding periods) - 1
• For investment A, this would be: 10.47% = (1 + (10% ÷ 12)) ^ 12 - 1
• And for investment B, it would be: 10.36% = (1 + (10.1% ÷ 2)) ^ 2 - 1

Investment B has a higher stated nominal interest rate, but the effective annual interest rate is lower than the effective rate for investment A. This is because Investment B compounds fewer times over the course of the year. If an investor were to put, say, $5 million into one of these investments, the wrong decision would cost more than $5,800 per year.

Special Considerations

As the number of compounding periods increases, so does the effective annual interest rate. Quarterly compounding produces higher returns than semiannual compounding, monthly compounding produces higher returns than quarterly, and daily compounding produces higher returns than monthly. Below is a breakdown of the results of these different compound periods with a 10% nominal interest rate:

• Semiannual = 10.250%
• Quarterly = 10.381%
• Monthly = 10.471%
• Daily = 10.516%

Limits to Compounding

There is a ceiling to the compounding phenomenon. Even if compounding occurs an infinite number of times—not just every second or microsecond, but continuously—the limit of compounding is reached.

With 10%, the continuously compounded effective annual interest rate is 10.517%. The continuous rate is calculated by raising the number “e” (approximately equal to 2.71828) to the power of the interest rate and subtracting one. In this example, it would be 2.171828 ^ (0.1) - 1.

How do you calculate the effective annual interest rate?

The effective annual interest rate is calculated using the following formula:

Effective Annual Interest Rate=(1+in)n −1where:i=Nominal interest raten=Number of periods \begin{aligned} &Effective\ Annual\ Interest\ Rate=\left ( 1+\frac{i}{n} \right )^n-1\\ &\textbf{where:}\\ &i=\text{Nominal interest rate}\\ &n=\text{Number of periods}\\ \end{aligned}​Effective Annual Interest Rate=(1+ni​)n−1where:i=Nominal interest raten=Number of periods​

Although it can be done by hand, most investors will use a financial calculator, spreadsheet, or online program. Moreover, investment websites and other financial resources regularly publish the effective annual interest rate of a loan or investment. This figure is also often included in the prospectus and marketing documents prepared by the security issuers.

What is a nominal interest rate?

A nominal interest rate does not take into account any fees or compounding of interest. It is often the rate that is stated by financial institutions.

What is compound interest?

Compound interest is calculated on the initial principal and also includes all of the accumulated interest from previous periods on a loan or deposit. The number of compounding periods makes a significant difference when calculating compound interest.

The Bottom Line

Banks and other financial institutions typically advertise their money market rates using the nominal interest rate, which does not take fees or compounding into account. The effective annual interest rate does take compounding into account and results in a higher rate than the nominal. The more the periods of compounding involved, the higher the ultimate effective interest rate will be.

The higher the effective annual interest rate is, the better it is for savers/investors, but worse for borrowers. When comparing interest rates on a deposit or a loan, consumers should pay attention to the effective annual interest rate and not the headline-grabbing nominal interest rate.

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