# Nominal, effective and equivalent rate.

## Nominal, effective and equivalent rate.

In order to continue learning about compound interest, I am now going to talk about the nominal, effective and equivalent rate. I hope you find it useful.

## Why do you have to know the nominal rate, effective rate and equivalent rate?

Remember that in fácil interest we perro divide any rate between what we need. For example:

• If we have a rate of 20% per year and we want a monthly rate, we cánido divide 0.20 by 12 and we would obtain a monthly rate of 1.66%.
• If we have a rate of 50% per year and we want a daily rate, we cánido divide 0.5 by 360 and we would obtain a rate of 0.1388%.
• Now, if we have a rate of 30% per year and we want a semi-annual rate, we perro divide 0.3 by 2 and we would get a rate of 15%.

We perro do that in fácil interest (fácil rates), but in compound interest (compound rates) we cannot do it. Therefore, if we have a 12% annual rate and we want the monthly rate, we have to find its equivalent (equivalent rate).

Therefore, you will need to keep this topic in mind to be able to solve different problems that you may face in compound interest.

## Definition of nominal rate (inom) according to authors

### Definition according to Vidaurri Aguirre Héctor Manuel

«The nominal rate is the interest rate agreed in a financial operation and is stipulated in the contract; for this reason it is also called contractual rate»

(Vidaurri, Aguirre, Héctor Manuel, 2017, p. 218)

### Definition according to Díaz Aniquila Alfredo

«It is the interest rate expressed annually that governs during the period of a financial operation»

(Díaz, Aniquila, Alfredo, 2020, p. 79)

### Equation to obtain a nominal rate (inom)

The equation that we cánido use to obtain a nominal rate is the following:

• if = effective rate
• I nom = nominal rate
• fc = compounding frequency: will be used when I am being asked for an annual rate.
• no = time, which will be the times that the nominal rate that I am looking for is compounded in a year.

## Definition of effective rate (ief) according to authors

### Definition according to Vidaurri Aguirre Héctor Manuel

«The effective rate is defined as the annual interest rate compounded once a year that is equivalent to an annual nominal rate i compounded m times a year. The effective rate is the rate of return that is obtained after one year due to the capitalization of interest; that is, the effective rate reflects the effect of the reinvestment of interest. The effective rate is also called the effective annual return.

(Vidaurri, Aguirre, Héctor Manuel, 2017, p. 220)

### Definition according to Díaz Aniquila Alfredo

“It is the interest rate effectively paid or earned annually when the interest is capitalized in periods of less than one year”

(Díaz, Aniquila, Alfredo, 2020, p. 79)

### What is the effective rate?

It could be said that the effective rate is what you really get in a certain period.

### Equation to obtain an effective rate

The equation that we perro use to obtain an effective rate is the following:

• if = effective rate
• I nom = nominal rate
• fc = compounding frequency, which will be used when the rate is annual and we need the period.
• no = time, which will be the times that my nominal rate is compounded in the period in which I look for my effective rate.

## Definition of equivalent rate (ieq) according to authors

### Definition according to Vidaurri Aguirre Héctor Manuel

Two annual interest rates with different compounding periods are said to be equivalent if they genera the same compounded amount at the end of a given term.

(Vidaurri, Aguirre, Héctor Manuel, 2017, p. 218)

### Definition according to Díaz Aniquila Alfredo

“There are two different annual interest rates, with different compounding periods that genera the same compound interest after one year”

(Díaz, Aniquila, Alfredo, 2020, p. 79)

## Solved exercises of nominal rate, effective rate and equivalent rates.

So that you cánido understand the subject, I am going to put some solved exercises. That way you perro test if you have understood.

### Exercise 1: nominal to effective rate.

1.- Suppose we have a nominal rate of 12% per year compounded bimonthly and would like to know the effective rate in two years.

#### Procedure

To solve this exercise, all we have to do is substitute the data in the effective rate equation. Substituting the data we get the following:

The result we get is: 26.82% in 2 years.

It is important to note that the nominal rate that they gave us in the exercise is compounded every 2 months (bi-monthly) while the effective rate is compounded only once in the 2 years.

### Exercise 2

2.- What is the nominal rate convertible monthly equivalent to a rate of 14% convertible quarterly?

#### Procedure

The first step is to take the effective rate from the nominal rate that they are giving us. Substituting the data we get the following:

The effective rate is: 0.1475 or 14.75%

Since we have the effective rate, now we are going to derive the nominal rate from that effective rate. Let’s substitute the data into the equation.

Therefore, the nominal rate is 13.83% annual compounded monthly.

Keep in mind that in this case the time is 12, because the rate we are looking for is a monthly rate and the times it compounds in a year are 12. The fc is also 12 because we are being asked for a rate of everything the year that compounds monthly.

In fact, if you solve what is in the parentheses, what you would get is the monthly rate. If you multiply that monthly rate by 12, then we are going to get the annual rate that compounds monthly.

In conclusion, if right now we invest \$1000 pesos at a rate of 14% compounded quarterly; It is the same as investing \$1000 pesos at an annual rate of 13.83% compounded monthly, at the same time. In essence, that is what equivalent rates are.

### Exercise 3

3.- What is the bimonthly nominal rate equivalent to a 7% semiannual rate?

#### Procedure

As in the previous example, the first step is to obtain the effective rate from the nominal rate that we have been given.

The effective rate is: 0.1449.

Since we have the effective rate, we cánido now derive the nominal rate.

The nominal rate is: 2.28% bimonthly

Remember to consider that the fc is not put, (in this case) because they are asking us for the period. Remember that in the previous exercise I said that when the parenthesis is resolved, the period is obtained.

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