Finance Notes Annuities Page 1 of 8 ANNUITIES Objectives: After completing this section, you should be able to do the following: • Calculate the future value of an ordinary annuity. • Calculate the amount of interest earned in an ordinary annuity. • Calculate the total contributions to an ordinary annuity. • Calculate monthly payments that will produce a given future value. Vocabulary: As you read, you should be looking for the following vocabulary words and their definitions: • ordinary annuity • simple annuity • Christmas club • tax-deferred annuity (TDA) • present value of an annuity Formulas: You should be looking for the following formulas as you read: • future value of an ordinary annuity • total contribution to an annuity • interest earned on an annuity • present value of an annuity An annuity is defined by merriam-webster.com as “a sum of money payable yearly or at other regular intervals”. Wikipedia defines an annuity as “any recurring periodic series of payment”. Some examples of annuities are regular payments into a savings account, monthly mortgage payments, regular insurance payments, etc. Annuities can be classified by when the payments are made. Annuities whose payments are made at the end of the period are called ordinary annuities. Annuities whose payments are made at the beginning of the period are called annuity-due. In this class we will only work with ordinary annuities. ordinary annuity annuity dueFinance Notes Annuities Page 2 of 8 We will need to be able to calculate the future value of our annuities. In order to do this we will need a formula to calculate future value if we know the amount of the payment, the interest rate and compounding period, and the number of payments. NOTE: The payment period and the compounding period will always match in our problems. Annuities which have the same payment and compounding period are called simple annuities. Example 1: Find the future value of an ordinary annuity with $150 monthly payments at 6¼% annual interest for 12 years. Solution: For this problem we are given payment amount ($150), the interest rate (.0625 in decimal form), the compounding period (monthly or 12 periods per year), and finally the time (12 years). We plug each of these into the appropriate spot in the formula −+=nrnrpymtFVtn11*. This will give us Future Value of an Ordinary Annuity −+=nrnrpymtFVtn11* FV = future value pymt = payment amount r = interest rate in decimal form n = number of compounding periods in one year t = time in years simple annuityFinance Notes Annuities Page 3 of 8 04651.32051120625.1120625.115012*12=−+=FVFV In this class we will round using standard rounding. This will make the future value $32051.05. A Christmas club account is a short-term special savings account usually set up at a bank or credit union in which a person can deposit regular payments for the purposes of saving money for Christmas purchases. Example 2: On March 9, Mike joined a Christmas club. His bank will automatically deduct $210 from his checking account at the end of each month, and deposit it into his Christmas club account, where it will earn %415 annual interest. The account comes to term on December 1. Find the following: a. Find the future value of Mike’s Christmas club account. b. Find Mike’s total contribution to the account. c. Find the total interest earned on the account. Solution: a. For this part we will use the future value formula for an ordinary annuity. The payment amount is 210. The interest rate in decimal form is .0525. The number of compounding period in one year is 12 (monthly payments). The amount of time in years (t) is 129. We will be making payments for 9 months (end of March, end of April, end of May, end of June, end of July, end of August, end of September, end of October, and finally end of November). This will give us Future Value of an Ordinary Annuity −+=nrnrpymtFVtn11* FV = future value pymt = payment amount r = interest rate in decimal form n = number of compounding periods in one year t = time in years Christmas clubFinance Notes Annuities Page 4 of 8 414866.1923120525.1120525.1210129*12=−+=FVFV In this class we will round using standard rounding. This will make the future value $1923.41. b. To find the total amount of Mike’s contribution, we only need to take the amount of each monthly payment and multiply by the number of payments per year and finally multiply by the number of years. This formula can be seen in the box on the left. This will give us 1890129*12*210 = . Thus the total contribution made by Mike is $1890. c. To find the interest earned by Mike, we need to use the interest formula to the left. This basically takes the total contribution and subtracts it from the future value. The difference between these two numbers will be the interest earned on the annuity. Plugging into the formula we get 40.33189041.1923129*12*21041.1923 =−=− . Thus Mike earns $33.40 in interest on this account. A tax-deferred annuity (TDA) is an annuity in which you do not pay taxes on the money deposited or on the interest earned until you start to withdraw the money from the annuity account. Example 3: John Jones recently set up a tax-deferred annuity to save for his retirement. He arranged to have $50 taken out of each of his biweekly checks; it will earn %838 annual interest. He just had his thirty-fifth Annuity Interest Formula tnpymtFVI**−= I = interest FV = future value pymt = payment amount n = number of compounding periods in one year t = time in years Annuity Total Contribution tnpymt**oncontributitotal= pymt = payment amount n = number of compounding periods in one year t = time in years tax-deferred annuityFinance Notes Annuities Page 5 of 8 birthday, and his ordinary annuity comes to term when he is sixty-five. Find the following: a. Find the future value of John’s annuity. b. Find John’s total contribution to the annuity. c. Find the total interest earned on the annuity. Solution: a. For this part we will use the future value formula for an ordinary annuity. The payment amount is 50. The interest rate in decimal form is .08375. The number of compounding