1. Annuity a sequence of payments, each made at equally spaced time intervals.

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1 Ordinary Annuities (Young: 6.2) In this Lecture: 1. More Terminology 2. Future Value of an Ordinary Annuity 3. The Ordinary Annuity Formula (Optional) 4. Present Value of an Ordinary Annuity More Terminology 1. Annuity a sequence of payments, each made at equally spaced time intervals. 2. Ordinary Annuity an annuity in which the interest on the previous payments made is compounded at the same time the new payment is made. Note: All the annuities in this class will be ordinary. Examples of Ordinary Annuities 1. A sequence of annual payments into an IRA (Individual Retirement Account); 2. A sequence of equal monthly payments to pay off a car loan; 3. A sequence of equal monthly payments to pay off a house loan; 4. A sequence of equal quarterly payments to save for tuition needed by a child in the future. Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 1

2 Future Value of an Ordinary Annuity Class Activity Suppose that we set up an ordinary annuity account, where payments of $1,000 each are placed in the account at the end of each year for four years and that the interest is 3%, compounded annually. How much will be in the account (the future value of the annuity) at the end of the four years? (A) To assist you in answering this question, complete the table below, which traces the accumulated value of each payment made over the four years Years $1000 $1000 $1000 $1000 Payment # Amount of Payment Number of 3% Accumulated value of Payment 1 $1,000 2 $1,000 3 $1,000 4 $1,000 Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 2

3 (B) Fill in the total accumulated value, A, of the account for all the payments, written as a sum and find the value of this sum. A = Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 3

4 Class Activity (Optional) (A) The sum in part (B) of the previous class activity, A = (1.03)+1000(1.03) (1.03) 3 is called a geometric series. What we would like is a simple expression that gives the value of this sum without actually having to add up all the terms (this is convenient when there are many terms to add up). To find this expression fill in each equation below, writing sums in each case A = A= (B) Now use the equations above to compute (cancel all the terms that you can as you do this) 1.03 A A = Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 4

5 (C) Solve this last equation for A and you obtain the desired expression representing the sum, A without actually having to add all the terms together. (D) Verify that the expression in (C) gives the same value for the sum as you calculated in part (B) of the previous class activity. Answers:(A) 1.03A = 1000(1.03) +1000(1.03) (1.03) (1.03) 4,!A =!1000!1000(1.03)!1000(1.03) 2!1000(1.03) 3 ; (B) 1.03A! A = 1000(1.03) 4!1000 ; (C) A = 1000 (1.03)4! Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 5

6 Future Value of an Ordinary Annuity Let p = amount of each payment made at the END of each period (small case p and not P (principal)) t = number of years m = number of periods (payments) per year r = annual interest rate, compounded m times per year, as a decimal i = r = interest rate per period m n = m!t= total number of periods (payments) in t years Then the future value of the annuity is the sum of all the payments and interest earned on those payments and is found by the formula, Remark A = p ( *! *# " * * * )* 1+ r m% r m $ mt + & 1, ( * = p (1+i)n 1 * i Solving the formula directly above for p using algebra gives, p = A! " r % $ # m& 1 + r = A! % mt (1 m& " $ # ) i (1 +i) n (1 The quantity, p, is the payment (a sinking fund payment) we must deposit each period to achieve the amount, A, after t years. Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 6 +,

7 Class Activity (Optional) Use the TVM Solver on your graphing calculator to find the amount in the previous class activity, which we repeat here: Suppose that we set up an ordinary annuity account, where payments of $1,000 each are placed in the account at the end of each year for four years and that the interest is 3%, compounded annually. How much will be in the account (the future value of the annuity) at the end of the four years? [Hints: since we are starting with $0, put PV = 0. Also, since payments are being made into the account enter the payment as a NEGATIVE number.] Answer: Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 7

8 Class Activity At the end of each year John invests $1,000 in a company retirement plan in which the employer matches the employee s contribution. The plan pays 8% compounded annually and John plans to retire in 30 years (A) Make a guess as to how much money John will have in his account after 30 years. (B) What will be the total accumulated value of the account after 30 years? Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 8

9 (C) Suppose John delays making payments so that now he has 20 years before retirement. What will be the total accumulated value of the account after 20 years? (D) Suppose that John has really procrastinated in getting his retirement savings on track and thus has just 10 years to retirement. What will be the total accumulated value of the account after 10 years? Remark Notice that even after 20 years John has earned less than half the amount that he will get if he can save for an additional 10 years. This is the reason behind the old saying that the sooner you start saving for retirement, the better. Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 9

10 Class Activity A freight company estimates that it will need a forklift in 6 years. The cost of the forklift is $40,000. The company sets up a sinking fund that pays 6% compounded semiannually, in which it will make semiannual payments to achieve the goal. Calculate the size of each payment. Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 10

11 The Ordinary Annuity Formula (Optional) Remark If an interestbearing account already has an amount P in it, and equal payments p are made into the account periodically, as in an ordinary annuity, then the amount in the account after n periods is given by, A= (Future value of principal, P) + (Future value of annuity) = P(1 + i) n + p $ (1 + i)n!1 $ i! = P 1 + r # " m $ mt & % " # (! *# * * * ) + p 1 + r *" m % & $ mt + & % 1, r m where, p = amount of each payment made at the END of each period (small case p and not P (principal)) t = number of years m = number of periods (payments) per year r = annual interest rate, compounded m times per year, as a decimal i = r = interest rate per period m n = m!t= total number of periods (payments) in t years Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 11

12 Class Activity (Optional) A freight company estimates that it will need a forklift in 6 years. The cost of the forklift is $40,000. The company sets up a sinking fund that pays 6% compounded semiannually, in which it will make semiannual payments to achieve the goal. Calculate the size of each payment, assuming that the company currently has $5,000 in the account. What payment will be needed now? Answer: $ Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 12

13 Present Value of an Ordinary Annuity The present value of an ordinary annuity is defined to be the single deposit that will finance the annuity. To find a formula for the present value, P, of an annuity, solve for P in the following equation: P(1+ i) n " = p (1+ i)n!1 $ i # $ % & Accumulated value of a single deposit of P dollars after n periods under compound interest Future value of the annuity after n periods with payments of p dollars each period Using algebra we can solve for P to get: " P = p $ (1 + i)n!1 $ i(1 + i) n # % & " = p$ 1! (1 + i) n $ i # % & Divide the numerator and denominator in the first expression by (1+ i) n to get this equivalent expression. Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 13

14 Present Value of an Ordinary Annuity Formula For an annuity with, p = amount of each payment made at the END of each period (small case p and not P) t = number of years m = number of periods (payments) per year r = annual interest rate, compounded m times per year, as a decimal i = r = interest rate per period m n = m!t= total number of periods (payments) in t years the present value of the annuity is: " P = p (1+ i)n!1 $ $ i(1+ i) n # % & " = p 1! (1+ i) n $ i # $ % & Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 14

15 Class Activity You just retired and are planning a trip. You will need $800 per month for the 8 months that you will be gone. How much should you invest in the account, which pays 6% annual interest, when you depart so that you can withdraw the desired $800 each month for 8 months? Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 15

16 Remark The table below shows how the lump sum of $ invested when you depart generates the $6400 (= 8 x 800) that will be withdrawn from the account. Month New Balance = * (Old balance) 800 # [ i = 0.06/12 =.005 (monthly rate)] (initial deposit) ( ) 800 = ( ) 800 = ( ) 800 = ( ) 800 = ( ) 800 = ( ) 800 = ( ) 800 = (796.02) 800! $0 Funds resulting from interest = $6,400 $ = $ You can check your answer using the TVM Solver, as we did below. Note that the payment is entered as a negative number and that we set FV = 0 when computing present value: The present value of the payments is $ Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 16

17 Remark Let s prove that the present value formula actually gives the correct lump sum payment, P, which will fund the withdrawals over the 8 months. Let P be the amount of money that we will deposit to fund our vacation. Then the balance in the account after: 1. one month is: P(1.005)! two months is: (P(1.005)! 800)1.005! three months is: = P(1.005) 2! 800(1.005)! 800! (P(1.005) 2! 800(1.005)! 800)1.005! 800 = P(1.005) 3! 800(1.005) 2! 800(1.005)! eight months is: P(1.005) 8! 800(1.005) 7! 800(1.005) 6!!! 800(1.005)! 800 But the balance after eight months will be zero, so P(1.005) 8! 800(1.005) 7! 800(1.005) 6!!! 800(1.005)! 800 = 0 Solving for P in this equation and simplifying yields: " P=800 1!(1.005)!8 % $!6, # & Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 17

18 Class Activity (Optional) You just purchased a new gourmet oven for your kitchen. You will make payments of $100 a month for 3 years. The company that you purchased your oven from has decided to sell your loan to a bank and to receive a lump sum now, instead of collecting all the payments from you. How much should the company receive from the bank for your loan? Assume that interest rates on loans of this type are currently 9% per year. Answer: Compute the present value of the payments and get $ Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 18

19 Remark You can check your work in the previous example using the TVM Solver. The values that were inputted are captured below. Notice that the payment is entered as a negative number and the future value is entered as zero (i.e., FV = 0). The present value is $3, to the nearest cent. Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 19

20 Young 6.2 Survival Guide Notes copyright 2008 Knobel/Stanley 20