# 5.1 Fundamentals of Annuities

## LEARNING OBJECTIVES

- Understand terminology associated with annuities.
- Classify annuities based on the frequency of the payments and the compounding frequency.

An **annuity **is a continuous stream of equal periodic payments from one party to another for a specified period of time to fulfill a financial obligation. An **annuity payment **is the dollar amount of the equal periodic payment in an annuity environment. The payments are continuous, equal, periodic, and occur over a fixed time frame. If any one of these four characteristics is not satisfied, then the financial transaction fails to meet the definition of a singular annuity and requires other techniques and formulas to solve.

## Annuity Terms

The **payment interval** is the interval of time between two consecutive payments. For example, if payments are made every month, then the payment interval is monthly and the time period between two successive payments is one month.

The **payment frequency **([latex]P/Y[/latex]) is the number of times payments are made every year. That is, the payment frequency is the number of payment periods in one year.

Payment Interval |
Number of Times per Year Payments are Made |
Payment Frequency |

Annually | Once a year | 1 |

Semi-annually | Twice a year/Every six months | 2 |

Quarterly | Four times a year/Every three months | 4 |

Monthly | Twelve times a year/Every month | 12 |

Semi-monthly | 24 times a year | 24 |

Bi-weekly | 26 times a year | 26 |

Weekly | 52 times a year | 52 |

Daily | Every day/365 times a year | 365 |

The **term or time period **([latex]t[/latex]) of an annuity is the length of time from the beginning of the first payment interval to the end of the last payment interval.

Annuity calculations require the **total number of payments during the term **([latex]n[/latex]). As with compound interest, to calculate the total number of payments, the time must be in years. If the term is not in years, the term must be converted to years. If the term is given in months, divide by 12 to convert the term to years. If the term is given in days, divide by 365 to convert the term to years.

[latex]\begin{eqnarray*} \mbox{Total Number of Payments} & = & \mbox{Payment Frequency} \times \mbox{Time in Years} \\ n & = & P/Y \times t \end{eqnarray*}[/latex]

## EXAMPLE

For each of the following, calculate the number of payments.

- $200 deposited every month for five years.
- $750 deposited every quarter for three years and nine months.
- $300 loan payments semi-annually for 54 months.

**Solution:**

- [latex]n = P/Y \times t = 12 \times 5 = 60[/latex]
- [latex]n = P/Y \times t = 4 \times \frac{45}{12} = 15[/latex]
- [latex]n = P/Y \times t = 2 \times \frac{54}{12} = 9[/latex]

### NOTE

Annuity calculations do **not** require the total number of compoundings and the calculation of [latex]n[/latex] for an annuity does not involve the compounding frequency.

It now becomes critical to ensure the proper application of the cash flow sign convention on the calculator—failure to do so will result in an incorrect answer. For example, if you borrow money and then make annuity payments on it, you enter the present value (PV) as a positive (you received the money) while you enter the annuity payments as negatives (you paid the money to the bank). This results in future balances getting smaller and you owing less money. If you inadvertently enter the annuity payment as a positive number, this would mean you are borrowing more money from the bank so your future balance would increase and you would owe more money. These two answers are very different!

## Types of Annuities

Annuities are classified in two ways—by the timing of the payment (at the end or beginning of the payment interval) and by whether or not the payment frequency and the compounding frequency are equal.

In an **ordinary annuity**, the payments are made at the end of the payment interval. The first payment occurs one interval after the beginning of the annuity and the last payment occurs on the same date as the end of the annuity. For example, if you took out a car loan today that requires you to make monthly payments to repay the loan, your first payment would occur one month from today at the end of the first payment interval and all of your subsequent payments would occur at the end of each month. Common applications of ordinary annuities include bank loans, mortgages, bonds, and Canada Pension Plan payments.

In an **annuity due**, the payments are made at the beginning of the payment interval. The first payment occurs on the same date as the beginning of the annuity and the last payment occurs one payment interval before the end of the annuity. For example, if you rent an apartment, your rent payment forms an annuity due because the your rent is paid at the beginning of every month. Common applications of annuities due include any kind of lease, any kind of rental, membership dues, and insurance payments.

In a **simple annuity**, the frequency of the payments and the compounding frequency for the interest rate are equal. For example, an annuity with quarterly payments and an interest rate that compounds quarterly is a simple annuity.

In a **general annuity**, the frequency of the payments and the compounding frequency for the interest rate are not equal. For example, an annuity with quarterly payments and an interest rate that compounds monthly is a general annuity.

Altogether, there are four types of annuities—ordinary simple annuity, ordinary general annuity, simple annuity due, general annuity due. The table below summarizes the four types of annuities and their characteristics.

Annuity Type |
Timing of Payments in aPayment Interval |
Payment Frequency and Compounding Frequency |
Start of Annuity and First Payment Same Date? |
End of Annuity and Last Payment Same Date? |

Ordinary Simple Annuity | End | Equal | No, first payment one interval later | Yes |

Ordinary General Annuity | End | Unequal | No, first payment one interval later | Yes |

Simple Annuity Due | Beginning | Equal | Yes | No, last payment one interval earlier |

General Annuity Due | Beginning | Unequal | Yes | No, last payment one interval earlier |

#### Attribution

“11.1: Fundamentals of Annuities” from Business Math: A Step-by-Step Handbook Abridged by Sanja Krajisnik; Carol Leppinen; and Jelena Loncar-Vines is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

“11.1: Fundamentals of Annuities” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.