Annuities

Annuities Explained: Types and Examples | CFA Level I

PREREQUISITE LESSON

This lesson is a prerequisite for the course. While you won’t be directly tested on its content in the exam, it’s assumed you’ve gained this knowledge or skill during your university studies. We strongly recommend reviewing this lesson, as its content may be essential for understanding subsequent parts of the curriculum.

In this lesson, we’ll explore annuities, an application of the time value of money concept for a series of cash flows. We’ll learn about ordinary annuities, annuity dues, and perpetuities. Let’s dive in!

What is an Annuity?

An annuity is a finite set of level sequential cash flows. It has three key features:

  • Finite: A predetermined number of cash flows.
  • Level: The same amount for every cash flow.
  • Sequential: Cash flows disbursed in fixed time periods.

There are two types of annuities:

  1. Annuity due: Cash flow occurs at the beginning of each period.
  2. Ordinary annuity: Cash flows occur at the end of each time period.

A perpetuity is like an ordinary annuity, but the cash flows never end.

EXAMPLE

Calculating Present Value of Annuities

Let’s examine an example where we need to find the present value of an annuity.

EXAMPLE

Takashi is 60 years old and has an ordinary annuity that will pay him $100,000 a year from age 65 to 75. He wants to sell the annuity today and use the proceeds to buy a house. Assuming a 4% discount rate, how much can he expect to gain from the sale?

PV at age 65 = 100,000 * [1 – (1 + 0.04)^(-10)] / 0.04 = $811,090

Discount PV65 to PV60: PV60 = PV65 / (1 + 0.04)^5 = $666,657

Perpetuities

The formula for a perpetuity is simple: take the payment amount per period and divide it by the interest or discount rate.

PVperp = PMT / r

EXAMPLE

What if Takashi’s annuity is a perpetuity that pays $100,000 each year from age 65 instead?

PV = PMT / r

PV65 = 100,000 / 0.04 = $2,500,000

Discount PV65 to PV60: PV60 = PV65 / (1 + 0.04)^5 = $2,055,000

Calculating Annuity Payments

EXAMPLE

Casey has a $20,000 ordinary annuity with an expected annual return of 7%. He is expected to live for another 20 years. How much can he withdraw at the end of each year, so that the annuity lasts him for the full 20 years?

The answer is $1,888. This example shows the versatility of the TVM calculator. As long as you have four parameters in the TVM set, you can find the fifth unknown parameter.

Recap

In this lesson, we’ve covered three types of annuities:

Most annuity problems can be solved using the TVM calculator. The only formula you need to remember for this lesson is the simple formula to calculate the PV of a perpetuity.

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