Skip to main content

Annuities Types

Introduction

An annuity is a series of equal payments made at regular intervals over a specified period. Annuities are commonly used in financial products such as retirement plans, insurance policies, and loans. They help in the calculation of regular payments or the future value of a series of payments over time.

There are different types of annuities based on when the payments are made or received, and each type is suited for different financial situations. The main types of annuities include:

  1. Ordinary Annuity
  2. Annuity Due
  3. Deferred Annuity
  4. Perpetuity

1. Ordinary Annuity

An ordinary annuity is a series of equal payments made at the end of each period, such as monthly, quarterly, or annually. It is the most common type of annuity, where the payments or receipts occur at the end of each payment period.

Formula for the Future Value of an Ordinary Annuity:

[ FV = P \times \frac{(1 + r)^n - 1}{r} ]

Where:

  • ( FV ) = Future value of the annuity
  • ( P ) = Payment amount per period
  • ( r ) = Interest rate per period
  • ( n ) = Total number of payments

Example of Ordinary Annuity

Problem: You decide to save $200 at the end of each month for 5 years in an account that earns an annual interest rate of 6%, compounded monthly. What will be the future value of this ordinary annuity?

Solution:

  • ( P = 200 ) (Monthly payment)
  • ( r = \frac{6%}{12} = 0.005 ) (Monthly interest rate)
  • ( n = 5 \times 12 = 60 ) (Total number of payments)

Using the formula:

[ FV = 200 \times \frac{(1 + 0.005)^{60} - 1}{0.005} ]

[ FV = 200 \times \frac{1.34885 - 1}{0.005} ]

[ FV \approx 200 \times 69.77 = 13,954 ]

Explanation: The future value of this ordinary annuity, where $200 is deposited monthly at the end of each month, will be approximately $13,954 after 5 years.

2. Annuity Due

An annuity due is a series of equal payments made at the beginning of each period. Because each payment is made at the start of the period, an annuity due accumulates more interest than an ordinary annuity over the same term.

Formula for the Future Value of an Annuity Due:

[ FV = P \times \frac{(1 + r)^n - 1}{r} \times (1 + r) ]

Where:

  • The additional ( (1 + r) ) factor accounts for the extra compounding period.

Example of Annuity Due

Problem: You invest $300 at the beginning of each quarter for 3 years in an account that pays an annual interest rate of 8%, compounded quarterly. What will be the future value of this annuity due?

Solution:

  • ( P = 300 ) (Quarterly payment)
  • ( r = \frac{8%}{4} = 0.02 ) (Quarterly interest rate)
  • ( n = 3 \times 4 = 12 ) (Total number of payments)

Using the formula:

[ FV = 300 \times \frac{(1 + 0.02)^{12} - 1}{0.02} \times (1 + 0.02) ]

[ FV = 300 \times \frac{1.26824 - 1}{0.02} \times 1.02 ]

[ FV \approx 300 \times 13.412 \approx 4,023.60 ]

Explanation: The future value of this annuity due, where $300 is deposited at the beginning of each quarter, will be approximately $4,023.60 after 3 years.

3. Deferred Annuity

A deferred annuity is an annuity in which the first payment is delayed until a specified future date. It allows the accumulation of interest over a deferred period before the payments start.

Example of Deferred Annuity

Problem: You plan to invest $500 annually for 10 years in a retirement account, but the payments will only begin 5 years from now. If the account earns an annual interest rate of 5%, what will be the amount accumulated at the end of 15 years?

Solution:

  • Step 1: Calculate the future value of the annuity after 10 years.
  • Step 2: Calculate the accumulated value for the next 5 years without additional payments.

  1. Future Value after 10 Years:

    [ FV = 500 \times \frac{(1 + 0.05)^{10} - 1}{0.05} ]

    [ FV \approx 500 \times 12.5779 = 6,288.95 ]

  2. Value Accumulated after 5 More Years:

    [ FV_{\text{final}} = 6,288.95 \times (1 + 0.05)^5 ]

    [ FV_{\text{final}} \approx 6,288.95 \times 1.276 = 8,022.45 ]

Explanation: After deferring the annuity for 5 years, the total amount accumulated will be approximately $8,022.45 after 15 years.

4. Perpetuity

A perpetuity is an annuity that continues indefinitely, with regular payments made forever. It is commonly used to calculate the value of assets that generate constant cash flows, such as stocks with fixed dividend payments.

Formula for the Present Value of a Perpetuity:

[ PV = \frac{P}{r} ]

Where:

  • ( P ) = Payment amount per period
  • ( r ) = Interest rate per period

Example of Perpetuity

Problem: A stock pays a constant annual dividend of $100. If the required rate of return is 4%, what is the present value of this perpetuity?

Solution:

  • ( P = 100 ) (Annual dividend)
  • ( r = 4% = 0.04 ) (Required rate of return)

Using the formula:

[ PV = \frac{100}{0.04} = 2,500 ]

Explanation: The present value of a perpetuity that pays $100 annually, with a required rate of return of 4%, is $2,500. This means an investor would be willing to pay $2,500 today to receive $100 per year indefinitely.

Back to top