# ANNUITIES. Ordinary Simple Annuities

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1 An annuity is a series of payments or withdrawals. ANNUITIES An Annuity can be either Simple or General Simple Annuities - Compounding periods and payment periods coincide. General Annuities - Compounding periods and payment periods can be different. Both Simple and General Annuities can either be Ordinary or Due. Ordinary Annuity - Payment is at the end of the interval. - i.e., this happens when you borrow money and are making repayments. Annuity Due - Payment is at the beginning of the payment interval. - i.e., this happens when you are saving money. Now we learn how to use the PMT button. Present Value of Ordinary Simple Annuities Ordinary Simple Annuities You wish to make withdrawals of \$2000 from an account at the end of each quarter for the next fifteen months. In order to do this you are going to put money into an account today that pays 5 percent compounded quarterly. You will then make the quarterly withdrawals, starting three months from now. What is the present value of this stream of payments? 2 nd P/Y 4 Enter CE/C = 2 nd XP/Y N 5 I/Y 2000 PMT CPT PV PV = Why is the present value less than the sum of the withdrawals? Why is the 2000 entered as a positive number in this case? Why was the fifteen months divided by 12 for the number of periods? As part of his child-care agreement, Steven must make payments of \$750 at the end of every three months for the next 42 months. How much must he deposit in an account today that pays 12.1 percent compounded quarterly if he is to be able to make all of the payments from this account? Business Math Annuities Page #1

2 Future Value of Ordinary Simple Annuities What is the future value of a series of payments of \$1500 made at the end of every six months for 12 years if the interest is 8.5 percent compounded semi-annually? Compounded semi-annually 12 years \$1500 every six months 8.5% 2 nd P/Y 2 Enter CE/C 12 2 nd XP/Y N 8.5 I/Y / PMT CPT FV FV = \$60, Why is the present value entered as zero? Why is the payment entered as negative? What is the sum of the payments? What is the amount in an account at the end of 6.5 years if payments of \$575 are made at the end of every three months and the interest is 7.2 percent compounded quarterly? Payments for Ordinary Simple Annuities Doris wants to save \$6000 for a trip in 18 months. How much must she deposit in an account at the end of each month for 18 months that pays 6.3 percent compounded monthly if she is to have \$6000 in the account when she has made the last payment? 6000 needed 18 months 6.3% compounded monthly 2 nd P/Y 12 Enter CE/C 18 N 6.3 I/Y 600 CPT PMT PMT = Business Math Annuities Page #2

3 Why is the present value zero? Why is the payment negative? Chia wants to be able to withdraw money from an account at the end of every six months for the next 54 months. She deposits \$ into her account, which pays 5.3 percent compounded semi-annually. How much will she be able to withdraw every 6 months if, when she makes the last withdrawal, there is no money left in the account? Term for Ordinary Simple Annuities Deposits of \$ are made at the end of each month into an account that pays 6.6 percent compounded monthly. After which payment will the amount in the account be \$18,575.55? Monthly deposits of \$ Compounded monthly 6.6% Future value of \$18, nd P/Y 12 Enter CE/C 6.6 I/Y / PMT 18, FV CPT N N = 48 How many years? Zarfana has \$ in an account that pays 12.4 percent compounded semi-annually. She wants to make withdrawals of \$1200 at the end of every six months. How many withdrawals of this account will she be able to make? Interest Rate for Ordinary Simple Annuities What interest rate, compounded quarterly, will yield a future value of \$17, if payments of \$900 are made at the end of every three months for four years? Compounded quarterly Future value of \$17, Quarterly payments of \$900 4 years Business Math Annuities Page #3

4 2 nd P/Y 4 Enter CE/C 4 2 nd XP/Y N / PMT 17, FV CPT I/Y I/Y = 9.1 Sownthari has \$ in an account that pays interest compounded quarterly. If she will be able to make withdrawals of \$725 at the end of every three months for 2.5 years, what nominal interest rate is being paid on the account? Simple Annuities Due Your financial calculator assumes that we are working with ordinary annuities until we tell it otherwise. An annuity due is when the payment is made at the beginning of the payment period. On your financial calculator. Note that there is now a little BGN on the screen. That s all there is to it. Now your calculator will make the necessary adjustments to account for the payments being made at the beginning of the payment period. Present Value for Simple Annuities Due France has borrowed some money that she is paying back with payments of \$900 at the beginning of every six months for the next nine years. If the interest being charged is 5.2 percent compounded semiannually, what is the value of the original loan? 9 years \$900 every six months, at the beginning of the period 5.2% Compounded semi-annually 2 nd P/Y 2 Enter CE/C 9 N 5.2 I/Y Business Math Annuities Page #4

5 900 + / PMT CPT PV PV = Why is the future value entered as zero? Tammy has agreed to help with the upkeep of the family cottage. She will pay \$3600 at the beginning of each year for the next seven years. If she can deposit money in an account at 9.4 percent compounded annually, how much should be deposited so the \$3600 payments can be made from this account? Future Value for Simple Annuities Due Samina has just started to rent her basement apartment to a tenant. She plans to deposit all of the rent cheques she receives at the beginning of the month for the first two years into an account she will leave for emergency house repairs. If she receives \$525 a month and the account pays 7.8 percent compounded monthly, how much money will there be in the account at the end of two years if she has not made any withdrawals? Compounded monthly 2 years \$525 per month 7.8% 2 nd P/Y 12 Enter CE/C 2 2 nd XP/Y N 7.8 I/Y / PMT CPT FV FV = 13, Regat is starting to save for a vacation. She deposits \$1250 into an account now and then makes another three deposits of the same amount one year apart. If the account pays 7.1 percent compounded annually, what will be the balance in Regat s account four years from now? Business Math Annuities Page #5

6 Payments for Simple Annuities Due Addy has calculated that if she has a balance now of \$16,314 in an account that pays 10.5 percent interest compounded monthly, she will have enough money to cover her rent for the next 1.5 years. If she pays rent at the beginning of the month for this period of time, what is her monthly rent? Compounded monthly 1.5 years 10.5% \$16,314 present value 2 nd P/Y 12 Enter CE/C nd XP/Y N 10.5 I/Y / PV CPT PMT PMT = Why is the future value entered as zero? Why is the present value entered as a negative? What payments made at the beginning of each year for six years will repay a loan of \$12,500 if the interest rate is 12.4 percent compounded annually? Term for Simple Annuities Due Find the number of withdrawals and the term for an annuity that has a present value of \$4,295.44, if the interest is 8.2 percent compounded monthly, and the withdrawals of \$135 are made at the beginning of each month. Monthly payments of \$135 Compounded monthly 8.2% Present value of \$4, Business Math Annuities Page #6

7 2 nd P/Y 12 Enter CE/C 8.2 I/Y /- PV 135 PMT CPT N N = 36 How many years? Why are we talking about payment, not deposits? Why is the future value zero? Find the number of withdrawals and the term for an annuity that has a present value of \$200,000 if the interest is 3.8 percent compounded monthly, and the payments of \$2000 are made at the beginning of each month. Interest for Simple Annuities Due Nicole purchased an annuity that cost \$5, This annuity will pay her \$675 at the beginning of every six months for the next 60 months. If the interest is being compounded semi-annually, what nominal rate of interest is she earning? Compounded semi-annually Present value of \$5, Payments of \$675 at the beginning of every 6 months 60 months 2 nd P/Y 2 Enter CE/C = 2 nd XP/Y N / PV 675 PMT CPT I/Y I/Y = 8 Nicole purchased an annuity that cost \$500,000. This annuity will pay her \$20,000 at the beginning of every six months for the next 15 years. If the interest is being compounded semi-annually, what rate of interest is being charged? Business Math Annuities Page #7

8 Name: ANSWERS Set: Ordinary Simple Annuities Present Value of Ordinary Simple Annuities #1 How much must Steven deposit? Future Value of Ordinary Simple Annuities #2 What amount will be in the account? Payments for Ordinary Simple Annuities #3 How much would she be able to withdraw every 6 months? Term for Ordinary Simple Annuities #4 How many withdrawals? Interest Rate for Ordinary Simple Annuities #5 What is the nominal rate of interest? Simple Annuities Due Present Value for Simple Annuities Due #6 How much should be deposited now? Future Value for Simple Annuities Due #7 What will the balance in Regat s account be? Payments for Simple Annuities Due #8 What must the payments be? Term for Simple Annuities Due #9 How many payments? How many years? Interest for Simple Annuities Due #10 What rate of interest is being charged? Business Math Annuities Page #8

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