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Wize University Introduction to Finance Textbook > Time Value of Money

Annuities

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Annuities

  • A series of payments or cash flows with the following characteristics
  • Equal value
  • Equal time between cash flow
  • Equal interest rate
  • Finite number of cash flows
  • Payments can take place at the end of the period (ordinary annuity) or at the beginning of the period (annuity-due).

Present Value of an Ordinary Annuity

  • Sum of the present value of the individual cash flows one period before the first payment.
  • Can be solved using a formula or with a financial calculator.
  • In both cases, you will need:
  • Value of the cash flow
  • Effective periodic rate
  • Number of cash flows


Present Value Formula

Financial Calculator Method

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Future Value of an Ordinary Annuity

  • Sum of the future value of the individual cash flows at the same period of time as the last cash flow.
  • Can be solved using a formula or with a financial calculator.
  • In both cases, you will need:
  • Value of the cash flow
  • Effective periodic rate
  • Number of cash flows



Future Value Formula

Financial Calculator Method


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Example: Ordinary Annuities

Joey is calculating the present value of his future federal GST rebates. He estimates that he will receive $300 in each quarter of a calendar year. Joey's bank account earns an APR of 1.25% that compounds monthly.

1. What is the present value of Joey's total future GST rebates over a seven-year period?
2. If Joey decides that he will leave everything he's received in the bank, what will be his account balance in seven years?

Practice: Ordinary Annuities

Richard is mulling over purchasing an investment for $200,000. The investment will pay him $5,000 every 6 months for 40 years.
Richard can earn 10% per year on his other investments compounded monthly, should he make this investment today?
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Annuity-Due

  • Annuities with payments at the beginning of the period.
  • For example, you will receive $100 per year for 6 years, starting today.


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Example: Annuity-Due

Carmen and Carrie are sisters. Carmen donates $30 to World Vision Canada at the beginning of each month. Carrie donates $30 to World Vision Canada at the end of each month. Their bank accounts both offer an effective annual interest rate of 1.25%. Calculate, separately, the present value of each sister’s donations over the next ten years.

Practice: Annuity-Due

You plan on retiring in 40 years and have decided you will begin saving $500 per month, starting today. Your investment account earns 8% interest compounded quarterly, and you will make 480 total deposits. How much will you have in your account when you retire?

Round the effective rate to at least 6 decimal places and your final answer to the nearest dollar.
Extra Practice
An investment promises to pay you $500 per year for 7 years, starting one year from today. The payments will increase to $800 for the next 5 years, and then decrease to $200 per year for the last 8 years. The discount rate is 8% per year.
What is the present value of this investment?
Paul will receive $100 per year starting one year from today, the payments will continue until the end of year 5. This cash stream can be described as which of the following (select all that apply)?
Sarah just turned 21 years old and has decided to begin saving money. She plans to save $8,000 at the end of each year for 10 years, but only starting 4 years from today. Sarah's account earns a return of 12% compounded weekly. Her brother Matthew is 32 years old and just deposited $x into his savings account and plans to deposit an equal amount each year for a total of 12 years. Matthew's account earns 11% compounded continuously.
How much is Matthew's annual payment if both siblings' investments have the same present value?
Sarah took out a $40,000 loan at a 7% annual interest rate to pay for college. She plans to pay off the loan in 10 years with equal monthly payments. How much will her monthly payments be?
Rachel took out a $50,000 loan at a 6% annual interest rate to start a business. She plans to pay off the loan in 5 years with equal quarterly payments. How much will her quarterly payments be?
Maria took out a $20,000 loan at a 5% annually compounded interest rate. How much will she have to pay back in total after 3 years if she makes monthly payments?
Jenna took out a $30,000 loan at a 4% annual interest rate to buy a car. She plans to pay off the loan in 4 years with equal monthly payments. How much will her monthly payments be?
Tina wants to go to Monaco in two years. She has $1,000 now and she estimates her vacation would cost $7,500. If the interest rate is 11% per year compounded monthly, she would have to save ______ every month for the next 24 months to go on the vacation.
What is the present value of an annuity-due with $50 payments each year for 10 years? Assume a discount rate of 12%.
What is the present value of an ordinary annuity with $50 payments each year for 10 years? Assume a discount rate of 12%.
How much will you save in 30 years if you deposit $600 per month for the first 10 years, then $750 per month for the following 10 years and $900 per month in the final 10 years? Assume your account earns 18% compounded monthly. Round your intermediate values to 3 decimal places and your final answer to 2 decimal places.
Which of the following best describes a cash flow of $100 per year for 10 years starting today?
You plan on retiring in 45 years and wish to save up enough money to withdraw $20,000 per month for 30 years. The first withdrawal will be the money after you retire. For the next 45 years, your retirement savings account will earn 8% compounded monthly but will then decrease to 6% compounded monthly after retirement. How much should you save at the end of each month for the next 45 years to reach your goal?