## Resolved years of deferred annuities

In this article, my objective is for you to learn what deferred annuities are and I think that the best way to do it is by solving exercises.

Therefore, I am going to share a equipo of solved exercises that will help you familiarize yourself with the subject.

## How to start?

The truth is that if you are learning what an annuity is for the first time, then I would not recommend that you continue reading the article.

This is because I am not going to spend much time explaining the basic concepts of annuities or the basic concepts of **fácil interest** and **compound interest**.

Now, if you already know about fácil and compound interest, then I recommend that you start learning about the **annuities due** and on the **due annuities**.

I say this, since it is the basis for you to learn to solve deferred annuity problems.

Next, I am going to leave you two articles in which I espectáculo resolved exercises of due and anticipated annuities.

By the way, I recommend that you read the articles in the following sequence.

## Why start with due and anticipated annuities?

If you have already read my article on past due annuities, then you already know that I talk about the classification of annuities there and you will know that an annuity falls into many categories at the same time.

For example, an annuity perro be:

**true.****Fácil.****anticipated.****Deferred.**

Therefore, you are going to find problems about deferred due annuities or problems about deferred early annuities.

In such a way that you will have to know its operation and its formulas.

## What are deferred annuities?

With respect to initiation, an annuity cánido be **immediate** either **deferred**.

**Immediate Annuity:**It is immediate when payments or collections have to be made in the period immediately following the execution of the contract or deal.**Deferred Annuity:**It is deferred when the collection or payment is postponed for a while.It is often known as

**Grace period**either**deferral period**.

**In such a way that in deferred annuities the start of collections (deposits) is postponed for one or more periods. **

For example, imagine that I sell you a product X on credit and I tell you that you start paying in three months.

Therefore, you will not start paying in the immediate period, but you will have a grace period.

In the grace period they cánido give you 1 month, two months or those that are agreed.

If you still have doubts, do not worry, since with the examples it will be clear to you.

## Deferred (Mature) Annuity Formulas

The following formulas will help you solve the different problems of deferred due annuities.

The equations are the following:

- Vad = current value
- a = annuity or rent.
- n = time or number of payments.
- y = grace period.
- i = interest rate.

## Resolved years of deferred annuities (overdue)

Now, we are going to use the previous equations to be able to solve some problems about deferred annuities due.

### Exercise 1

**1.** A department store makes the well-known promotion: **“Buy now and pay later”**This offer makes you buy a computer on October 1, for which you will have to pay 500 pesos a month for 12 months.

Consider that you will have to start paying as of January 1, 2023 and that the interest rate is 30% convertible monthly.

What is the cash value of the computer?

The data we have to solve the exercise are the following:

to = | 500 |

n = | 12 |

i = | 0.30/12 |

and = | 2 |

Vad (current value) | ? |

### First Way to Solve Deferred Due Annuity Problems

Since they are asking us for the capital or also called present value, we are going to use the present value equation that I gave you previously.

Substituting the data into the elabora we get the following:

Therefore, one way in which you perro solve past due deferred annuity problems is with the help of the previous equations.

### Second way to solve deferred annuity problems

Deferred annuity problems, whether due or anticipated, perro be solved with the same equations or formulas with which annuity problems due or anticipated (immediate annuities) are solved.

However, we also have to make use of the compound interest equations to be able to add or subtract the interest corresponding to the grace period.

In such a way that the grace period is the only thing that makes the difference between deferred annuities and immediate annuities (habitual early and expired annuities).

In order to demonstrate what I just said, I am going to solve the previous problem with the capital elabora of the annuities due and with the compound interest capital equation.

**Let’s start!**

#### Step 1 to solve the past due deferred annuity problem in the second way:

We are going to substitute the data we have in the capital equation pertaining to the annuities due.

The equation is the following:

If you want to see all the expired annuity formulas, then clic on the following button:

#### Step 2

Well, now we are going to substitute the data that we have in the equation.

Doing so we get the following:

If it were a habitual (immediate) due annuity problem, then that would be our answer, but remember that there is a grace period with deferred annuities.

In this case, they give us a grace period of 2 months.

Therefore, we have to return the amount we obtained in 2 months time.

So what we’re going to do is use the primordial equation that you saw in compound interest and we’re going to go back in time to the amount that we got.

The equation to obtain the capital in compound interest is the following:

If we substitute the data in the above equation, we get the following:

As you cánido see, the result you get is the same, but in the second way you have to do an plus step.

Of course, the advantage of the second way is that you no longer have to memorize any more equations, since they are equations that you have already seen.

### Exercise 2

**2.** 10,000 are deposited in an investment fund in order to withdraw 4 monthly payments due as of January 1, 2024.

Consider that the grace period is 12 months and that the interest rate is 10% convertible monthly, **What is the value of the monthly payments that perro be withdrawn?**

The data we have is the following:

C = | 10000 |

n = | 4 |

i = | 15% convertible monthly |

and = | 12 |

to = | ? |

In this case, I am only going to solve the problem in the first way, but you cánido practice solving the problems with both ways.

So you perro see that the same result comes out and you cánido select the option that is easiest for you.

The equation that we are going to use to obtain the annuity or income in deferred annuities due is the following:

If we substitute the data, we get the following:

To solve the exercise with the annuities due formulas, then you have to use the income elabora.

However, first you have to take the 10,000 12 months into the future with the compound interest amount equation.

Then, the result you get you have to use as your current value (capital).

With that you will be able to obtain the same result.

### Exercise 3

**3. **

Today 1000 are deposited in an investment account that pays 10% compounded monthly.

How many monthly withdrawals due of 100 perro be made starting in 3 months?

The data we have is the following:

Vad = | 1000 |

n = | ? |

i = | 10% compounded monthly |

and = | 3 months |

to = | 100 |

The equation that we are going to need to solve this problem is that of time in annuities deferred overdue The equation is the following:

Now all that remains is to substitute the data in the equation and we will obtain the result.

## Deferred Annuity Due Formulas

Now I am going to give you the equations of the deferred annuities so that you cánido solve the problems.

The equations are the following:

In this case, Vd is equal to the current value.

From then on, almost all the equations only added a -1 to the grace period.

## Resolved years of anticipated deferred annuities

Now I am going to put some solved exercises of early deferred annuities.

### Exercise 1

**1.** Find the present value of a semi-annual rent of $10,000 for 5 years, if the first semi-annual advance payment is made 2 years from now.

Consider that the interest is 15% semiannually.

The data we have is the following:

to = | 10,000 semesters |

n = | 5 years or 10 semesters |

and = | 2 years or 4 semesters |

i = | 15% semester |

you = | ? |

#### Way 1 to answer the problem

The first way that we perro solve the above problem is with the help of the equations that I gave you before.

The equation that we are going to use is that of the current value (Vd).

Therefore, if we substitute the data we obtain the following:

Therefore, the present value is equal to **32999.21838**

#### Way 2 to answer the problem

As with deferred annuity due, we cánido also solve the problems using the annuity due formulas and the compound interest equations.

I will solve the problem in this way below.

**1. **

**
**

First, we are going to use the equation to find the primordial or present value in annuities due (immediate annuities).

The equation is the following:

Now we have to substitute the data and we get the following:

**2.** Now, we have to use the capital equation that is used in compound interest and we obtain the following:

As you cánido see, we get the same result as with form 1.

### Exercise 2

**2. **

You want to purchase a new computer on credit to use in your en línea classes.

The computer has a cash value of 25,000 and if it is acquired on credit, 12 monthly payments must be made in advance, the first of which must be made 3 months after the acquisition.

If the interest applied is 5% per month, **How much will you have to make monthly payments? **

The data we have is the following:

Vd (current value) = | 25000 |

n = | 12 monthly payments |

and = | 3 months |

i = | 5% or 0.05 monthly |

to = | ? |

To solve the problem, what we have to do is use the annuity or income equation (a) that I gave you before.

The equation is the following:

If we substitute the data we get the following:

### Exercise 3

**3.** You want to make an investment that allows you to dispose of 500,000 at the beginning of each year, for 3 years, and you want to receive the first annuity from the fourth year from the initial date of the investment.

Consider an interest rate of 25% per year.

**How much will you have to invest? **

Vd (current value) = | ? |

n = | 3 years |

and = | 4 years |

i = | 25% per year |

to = | 500000 |

If we substitute the data in the elabora we get the following:

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