# 2.1 Fundamentals of Annuities

## Learning Objectives

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

## Formula & Symbol Hub

#### Symbols Used

• $n$ or $N$ = Total number of annuity payments
• $P/Y$ = Number of payments per year or payment frequency
• $t$ = Time in years

#### Formulas Used

• ##### Formula 2.1 – Total Number of Payments (Annuity)

$n=P/Y \times t$

## Introduction

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 ($P/Y$) 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 ($t$) 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 ($n$). 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.

### $\boxed{2.1}$ Total Number of Payments (Annuity)

$\Large{\color{red}{n}}={\color{blue}{P/Y}}\times{\color{green}{t}}$

${\color{red}{n}}\;\text{ is Total Number of Payments:}$ total number of payments during the term.

${\color{blue}{P/Y}}\;\text{is Payment Frequency:}$ the number of times payments are made every year.

${\color{green}{t}}\;\text{is Time in Years:}$ the length of time from the beginning of the first payment interval to the end of the last payment interval.

### Example 2.1.1

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

1. $\200$ deposited every month for five years.
2. $\750$ deposited every quarter for three years and nine months.
3. $\300$ loan payments semi-annually for 54 months.
Solution
1. $n = P/Y \times t = 12 \times 5 = 60$
2. $n = P/Y \times t = 4 \times \frac{45}{12} = 15$
3. $n = P/Y \times t = 2 \times \frac{54}{12} = 9$

### Things to Watch Out For

Annuity calculations do not require the total number of compoundings and the calculation of $n$ 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 a Payment 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