Fundamental of Annuities

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.

There are four types of annuities, which are based on the combination of two key characteristics: timing of payments and frequency.

Four Types of Annuities

Ordinary Simple Annuity

An ordinary simple annuity has the following characteristics:

• Payments are made at the end of the payment intervals, and the payment and compounding frequencies are equal.
• The first payment occurs one interval after the beginning of the annuity.
• The last payment occurs on the same date as the end of the annuity.

Ordinary General Annuity

An ordinary general annuity has the following characteristics:

• Payments are made at the end of the payment intervals, and the payment and compounding frequencies are unequal.
• The first payment occurs one interval after the beginning of the annuity.
• The last payment occurs on the same date as the end of the annuity.

Simple Annuity Due

A simple annuity due has the following characteristics:

• Payments are made at the beginning of the payment intervals, and the payment and compounding frequencies are equal.
• The first payment occurs on the same date as the beginning of the annuity.
• The last payment occurs one payment interval before the end of the annuity.

General Annuity Due

A general annuity due has the following characteristics:

• Payments are made at the beginning of the payment intervals, and the payment and compounding frequencies are unequal.
• The first payment occurs on the same date as the beginning of the annuity.
• The last payment occurs one payment interval before the end of the annuity.

The table below summarizes the four types of annuities and their characteristics for easy reference.

Table 11.1.1. 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

One of the most challenging aspects of annuities is recognizing whether the annuity you are working with is ordinary or due. This distinction plays a critical role in formula selection later in this chapter. To help you recognize the difference, the table below summarizes some key words along with common applications in which the annuity may appear.

Table 11.1.2. Ordinary Versus Due Annuities
Type Key Words or Phrases Common Applications
Ordinary -… payments are at the end….
-… payments do not start today…
-… payments are later…
-… first payment next interval…
– bank loans of any type
– mortgages
– bonds
– Canada Pension Plan (CPP)
Due -…payments are at the beginning…
-…payments start today…
-… payments are in advance…
-…first payment today…
-…payments start now…
– any kind of lease
– any kind of rental
– RRSPs (usually)
– membership dues
– insurance

Annuities versus Single Payments

To go from single payments in Chapter 9 to annuities in this chapter, you need to make several adaptations:

Annuity Payment Amount (PMT). Annuity calculations require you to tie a value to this variable in the formulas and when you use technology such as the BAII+ calculator.

Payment Frequency or Payments per Year (P/Y). When you work with annuities, an actual value for P/Y is determined by the payment frequency. For simple annuities P/Y remains the same as C/Y, whereas the variables are different for general annuities.

Cash Flow Sign Convention on the Calculator. 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!

Definition and Computation of n. When you worked with single payments, $n$ was defined as the total number of compounds throughout the term of the financial transaction. When you work with annuities, $n$ is defined as the total number of payments throughout the term of the annuity. You calculate it using Formula 11.1 below.

The Formula

$\colorbox{LightGray}{Formula 11.1}\; \color{BlueViolet}{\text{Number of Annuity Payments:}\; n=P/Y \times \text{Number of Years}}$

where,

n is the total number of annuity payments.
P/Y is the number of payments per year.

How It Works

On a two-year loan with monthly payments and semi-annual compounding, the payment frequency is monthly, or 12 times per year. With a term of two years, that makes n = 2 × 12 = 24 payments. Note that the calculation of n for an annuity does not involve the compounding frequency.

Adapting Timelines to Incorporate Annuities

A good annuity timeline should illustrate the present value (PV), future value (FV), number of annuity payments (n), nominal interest rate (I/Y), compounding frequency (C/Y), annuity payment (PMT), and the payment frequency (P/Y). One of these variables will be the unknown.