The Compound Amount : If P dollars are deposited for n compounding periods at a rate of interest i per period, the compound amount A is A P 1 i n Example 1: Suppose $1000 is deposited for 6 years in an account paying 8% per year compounded annually. Then the compound amount is A P 1 i n 1000 1 0.08 6 1586.87 Let me introduce the variables in the definition. A is the future value P is the present value i is the interest rate per period n is the number of times that the amount is compounded. Example 2: Suppose that the value of a necktie that belonged to Elvis is $800 in the year 2002 and that the value appreciates by 10 percent each year thereafter. Then the value in the year 2003 is $880. Why? A P 1 i n 800 1.10 1 880.00 In the next year the value will increase by 10% or by $88. The value in the year 2004 will be A P 1 i n 800 1.10 2 968.00 Then the future value of the necktie is A P 1 i n in the year 2002 n 1
year n P 1 i n 2002 0 800 2003 1 880 2004 2 968 2005 3 2006 4 2007 5 You should complete the chart. Place into the Y screen the formula : Y1 800(1.10)^X and check out the table. Hence if the year is 2007, and n is 5, we obtain that the value of the necktie is: P 1 i n 800 1.10 n 1288.40 Example 3 The future value of $7500 invested in a savings account at 10% compounded quarterly for 9 years can be found as follows: A P 1 i n 7500 1 0.025 36 18244.01 Here we reason that the 10% annual rate is applied four times per year at the quarterly rate of 2.5% and there are 9 X 4 36 quarters in 9 years. Example 4 Find the amount of interest earned on an original investment of $43,000 at 10% compounded semiannually for 9 years. First compute the future value: A P 1 i n 43000 1 0.05 18 103484.63 2
The interest earned is the difference between the Present Value and the Future Value or $60,484.63 Examples (some with answers) 1. Vu Le deposited $18,000 in an account paying 12% compounded quarterly. How much interest will he earn in 7 years? $23,182.70 2. Ann Fridl-Marshall placed $25,000 in an account paying 10% compounded semiannually. How much interest will she earn in 5 years? 3. Electric rates are rising by 4% per year. If the average monthly bill is now $45, what will be the average monthly bill in 10 years? $66.61 4. The population of a certain state is now 12 million and is increasing at an annual rate of 6%. What will the population of the state be in 5 years? In 10 years? 5. The average price of a house nationally is now $78,000. Housing prices are increasing at a rate of 7% per year. What will the average price of a house be in 12 years? $176, 000 6. Annual tuition at a public university is now $2200 per year and is expected to increase at an annual rate of 8%. Room and board now costs $3600 per year and is expected to increase at an annual rate of 4%. What will be the total cost per year (tuition, room and board) in 15 years? 7. One bank pays 8% compounded semiannually; another pays 8% compounded quarterly. Find the difference in the interest earned in 5 years on a deposit of $24,000. $136. 87 8. Which is a more profitable rate for your investment: 12.5% compounded daily, 12.8% compounded monthly, or 13.5% compounded annually? 9. Luisa Gonzales needs $18,000 in 3 years. (a) What amount can Gonzales deposit today, at 8% compounded annually, so that she will have the needed money? $14,288.98 (b) How much interest will be earned? $3711. 02 (c) Suppose Gonzales can deposit only $12,000 today. How much would she be short of the $18,000 in 3 years? $2883.46 10. Brent Alderman must pay $27,000 in settlement of an obligation in 3 years. (a) What amount can he deposit today, at 12% compounded monthly, to have enough? (b) How much interest will he earn? (c) Suppose Alderman can deposit only $12,000 today. How much would he be short of the needed $27,000 in 3 years? 11. A movie theater ticket now costs $6. If prices continue to rise at 8% per year, approximately how many years will it take for the price to reach $1O? about 6.6 years 12. A pair of designer jeans now costs $45 and the price is expected to increase 12% each year. Approximately how many years from now will the price be $100? 3
13. Use the calculator to determine approximately how long it will take the general level of prices to double if the average annual inflation rate is (a) 4%; about 17. 7 years (b) 6%; about 11. 9 years (c) 8%; about 9. 0 years (d) 12%; about 6. 1 years 14. Round off your answers in Exercise 13 to the nearest year and compare them with the numbers 72/4, 72/6, 72/8, 72/12. Use this evidence to state a rule of thumb for determining approximate doubling time. This rule, which has long been used by bankers, is called the rule of 72. 15. Electricity Consumption (a) The consumption of electricity has increased historically at 6% per year. If it continues to increase at this rate indefinitely, find the number of years before the electric utilities would need to double their generating capacity. about 12 years (b) Suppose a conservation campaign coupled with higher rates caused the demand for electricity to increase at only 2% per year, as it has recently. Find the number of years before the utilities would need to double generating capacity. about 35 years Annuities ORDINARY ANNUITIES A sequence of equal payments made at equal periods of time is called an annuity. If the payments are made at the end of the time period, and if the frequency of payments is the same as the frequency of compounding, the annuity is called an ordinary annuity. The time between payments is the payment period, and the time from the beginning of the first payment period to the end of the last period is called the term of the annuity. The future value of the annuity, the final sum on deposit, is defined as the sum of the compound amounts of all the payments, compounded to the end of the term. For example, suppose $1500 is deposited at the end of the year for the next 6 years in an account paying 8% per year compounded annually. The table below shows this annuity schematically. 4
End of Year Deposit Value at the end of the 6 year term 2002 1500 1500.00 1 0.08 5 2203. 99 2003 1500 1500 1 0.08 4 2040. 73 2004 1500 1500 1 0.08 3 1889.57 2005 1500 1500 1 0.08 2 1749.6 2006 1500 1500 1 0.08 1 1620 2007 1500 1500 1 0.08 0 1500 To find the future value of this annuity, look separately at each of the $1500 payments. The first of these payments will produce a compound amount of 1500 1.08 5 Use 5 as the exponent instead of 6 since the money is deposited at the end of the first year and earns interest for only 5 years. The second payment of $1500 will produce a compound amount of 1500 1.08 4 The future value of the annuity is 1500 1.08 5 1500 1.08 4 1500 1.08 3 1500 1.08 2 1500 1.08 1 1500. (The last payment earns no interest at all. If you read this in reverse order, you see that it is just the sum of the first six terms of a geometric sequence with a 1500, r 1.08, and n 5. Therefore, the sum is 1500 1.086 1 1.08 1 11003.89 Example 5 Ted Nurdly is an athlete who feels that his playing career will last 7 years. To prepare for his future, he deposits $22,000 at the end of each year for 7 years in an account that pays 6% compounded annually. How much will he have on deposit after 7 years? End of Year Deposit Value at the end of the 7 year term 1 22000 22000.00 1 0.06 6 31207.42 2 22000 22000.00 1 0.06 5 29440.96 3 22000 22000.00 1 0.06 4 27774.49 4 22000 22000.00 1 0.06 3 26202.35 5 22000 22000.00 1 0.06 2 24719.20 6 22000 22000.00 1 0.06 1 23320.00 7 22000 22000.00 1 0.06 0 22000.00 5
The future value of the annuity is 22000 1.06 6 22000 1.06 5 22000 1.06 4 22000 1.06 3 22000 1.06 2 22000 1.06 1 22000 (The last payment earns no interest at all. If you read this in reverse order, you see that it is just the sum of the first seven terms of a geometric sequence with a 22000, r 1.06, and n 6. Therefore, the sum is 22000 1.067 1 1.06 1 184664.43 Examples (Some with answers) 1) Charlie is saving for a house. At the end of each year for 9 years he deposits $12,000 into a savings account that pays interest at 8% annually. What is the value of the annuity at the end of the 9 years? Answer: $149,850.69 Why? 12000 1.08 9 1 1.08 1 1. 49850.69 2) A 45-year old man puts $1000 in a retirement account at the end of each quarter until he reaches the age of 60 and makes no further deposits. If the account pays 11% interest compounded quarterly, how much will be in the account when the man retires at age 65? Answer : $256,015.48 The quarterly rate is 0.11 4 0.0275 So, in 15 years of regular quarterly deposits, the value of the annuity is 1000 1.0275 60 1 1.0275 1 $148809.14 This become the Present value P in the expression: A P 1 r x 6
With no further deposits for the last 5 years of his working life, the value of the account grows to 148809.14 1.0275 20 $256015.48 Sinking Funds The following is a repeat of the definition of the ordinary annuity presented earlier. ORDINARY ANNUITIES A sequence of equal payments made at equal periods of time is called an annuity. If the payments are made at the end of the time period, and if the frequency of payments is the same as the frequency of compounding, the annuity is called an ordinary annuity. The time between payments is the payment period, and the time from the beginning of the first payment period to the end of the last period is called the term of the annuity. The future value of the annuity, the final sum on deposit, is defined as the sum of the compound amounts of all the payments, compounded to the end of the term. The future value of a dollars paid at the end of each regular period for n 1 periods is given by the formula where r 1 i, and i is the interest rate applied each period. The formula for the future value of an annuity can be used to find the values of variables other than S. In the example below, the amount of money wanted at the end, S, is given, and we need to find a, the amount of each payment. EXAMPLE 1 Betsy Martini wants to buy an expensive video camera 3 years from now. She plans to deposit an equal amount at the end of each quarter for 3 years in order to accumulate enough money to pay for the camera. The camera will cost $2400, and the bank pays 6% interest compounded quarterly. Find the amount of each of the 12 deposits she will make. This example describes an ordinary annuity with S 2400, i.015 6%/4 1.5%, and n 1 3 4 12 periods. The unknown here is the amount of each payment, a. By the formula for the amount of an annuity, 2400 a 1.01512 1 1.015 1, Solution is: a 184. 03 The annuity in Example 1 is a sinking fund: a fund set up to receive periodic payments. These periodic payments ($184.03 in the example), together with the 7
interest earned by the payments, are designed to produce a given sum at some time in the future. As another example, a sinking fund might be set up to receive money that will be needed to payoff the principal on a loan at some time in the future. EXAMPLE 2 The Stockdales are close to retirement. They agree to sell an antique urn to the local museum for $17,000. Their tax adviser suggests that they defer receipt of this money until they retire, 5 years in the future. (At that time, they might well be in a lower tax bracket.) Find the amount of each payment the museum must make into a sinking fund so that it will have the necessary $17,000 in 5 years. Assume that the museum can earn 8% compounded annually on its money. Also, assume that the payments are made annually. These payments are the periodic payments into an ordinary annuity. The annuity will amount to $17,000 in 5 years at 8% compounded annually. Using the formula and a calculator, 17000 a 1.085 1 1.08 1, Solution is: a 2897. 76 Examples (Some with Answers) 3) Ray Berkowitz needs $10,000 in 8 years. (a) What amount should he deposit at the end of each quarter at 8% compounded quarterly so that he will have his $10,000? 10000 a 1.0232 1 1.02 1, Solution is: a 226. 11 (b) Find Berkowitz s quarterly deposit if the money is deposited at 6% compounded quarterly. 10000 a 1.01532 1 1.015 1, Solution is: a 245. 77 4) Harv s Meats knows that it must buy a new deboner machine in 4 years. The machine costs $12,000. In order to accumulate enough money to pay for the machine, Harv decides to deposit a sum of money at the end of each 6 months in an account paying 6% compounded semiannually. How much should each payment be? 8
5) Barb Silverman wants to buy an $18,000 car in 6 years. How much money must she deposit at the end of each quarter in an account paying 12% compounded quarterly so that she will have enough to pay for the car? 18000 a 1.0324 1 1.03 1, Solution is: a 522. 85 9