What nominal rate compounded monthly could be used instead of 10% compounded semi monthly

Answer:

An annual rate of 12% compounded monthly is 1% per month. A monthly rate of 1% is 1.01^3 = 1.030301 or 3.0301% quarterly. For a full year that comes to 1.01^12 = 1.12682503013197 or 12.682503013197%. Expressed as an annual rate compounded quarterly it’s 4*3.0301 = 12.1204%.

Video Transcript

So in this problem, we are given that we are we have an interest rate of 12% compounded monthly. We want to know the nominal rate if compounded quarterly. So the formula for that Is to do one plus the interest rate, Which is our 12 compounded monthly. We want to do it on a quarterly basis. Well there's three months and a quarter. Right? So we take this to the 1/3 and then we subtracted one off again. We do that, we get 3.85 percent is the I'm all right, A phenomenal rate compounded quarterly. That's equivalent to that. 12% compounded monthly. Okay. Then, were asked for the effective, effective, right, well, effective rate for that. You take e To this rate that we just got 0.0385. That's a track one off of that And he gets 0.0393. So that's 3.93 percent. Is the effective, Right? That is occurring well, here we go.

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

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 in order 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 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%

The 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.

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