# Business Math Chapter 13 Compound Interest Future Value

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Business Math Chapter 13: Compound Interest, Future Value and Present Value 1

13. 1 Compound Interest and Future Value n Find the future value by compounding manually. n Find the future value by using a $1. 00 future value table. n Find the effective interest rate. n Find the interest compounded daily using a table. 2 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Simple and Compound Interest n In some loans, interest is computed once during the life of the loan, using the simple interest formula. n In other loans, interest is computed more than once during the life of the loan or investment. n The interest is added to the principal and that amount becomes the principal for the next calculation of interest. n This process is called compounding interest. 3 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Key Terms n Interest period: the amount of time which interest is calculated and added to the principal. n Compound interest: the total interest that accumulated after more than one interest period. n Future value, maturity value, compound amount: the accumulated principal and interest after one or more interest periods. n Period interest rate: the rate for calculating interest for one interest period-the annual interest rate is divided by the number of periods per year. 4 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

13. 1 Find the future value by compounding manually n Dividing the annual interest rate by the annual number of interest periods gives us the period interest rate. n 12% annual interest rate divided by 2 interest periods yields a period interest rate of 6%, for example. n Using I= P x R x T, we can calculate the interest period, simplifying the formula to I = P x R, since T is one period. 5 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Find the period interest rate Period interest rate = Annual interest rate Number of interest periods per year 6 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Find the period interest rate for: n A 12% annual interest rate with 4 interest periods per year. n 3% n An 18% annual rate with 12 interest periods per year. n 1½% n An 8% annual rate with 4 interest periods per year. n 2% 7 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Find the future value Using the simple interest formula method: 1. Find the end of period principal: multiply the original principal by the sum of 1 and the period interest rate. 2. For each remaining period in turn, find the next end of period principal: multiply by the previous end of period principal by the sum of 1 and the period interest rate. 3. Identify the last end-of-period principal as the future value. 8 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Look at this example Find the future value of a loan of $800 at 13% for three years. n The period interest rate is 13% since it is calculated annually. n First end-of-year = $800 x 1. 13 = $904 n Second end-of-year =$904 x 1. 13 = $1021. 52 n Third end-of-year = $1021. 52 x 1. 13 = $1, 154. 32 n The FV of this loan is $1, 154. 32 9 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Find the compound interest n Compound interest = future value – original principal. n In the previous example, the compound interest is equal to the future value – original principal. n CI = $1, 154. 32 - $800 = $354. 32 n The compound interest = $354. 32 10 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Compare the compound interest amount to the simple interest n CI = $354. 32 n Simple interest for the same loan would be: n I = PRT n I = $800 x 0. 13 x 3 = $312. 00 n Simple interest would be $312. 00 n The difference between compound interest and simple interest for this loan = $354. 32 - $312 n The difference is $43. 32. 11 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Find the FV of an investment n Principal = $10, 000 n 8% annual interest rate, compounded semiannually n Find the FV at the end of three years. n Find the period interest rate: 8% ÷ 2 = 4% n Determine number of periods: 3 x 2 = 6 n Calculate each end-of-period principal. n Period 1 = 10, 000 x 1. 04 = $10, 400 12 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Find the FV of an investment n Second end-of-period principal = $10, 400 x 1. 04 = $10, 816 n Calculate each end-of-principal through the sixth end-of-period principal. n What is the final end-of-principal amount? n $12, 653. 19 13 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

13. 1. 2 Using a $1. 00 FV table n Since it would be tedious and time-consuming to calculate a large number of periods with the previous method, we can use Table 13 -1, which is the future value or compound amount of $1. 00. n Find the number of periods and the rate period to identify the value by which the principal is multiplied. 14 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Look at this example n Using Table 13 -1, find the compound interest on $500 for six years compounded annually at 8%. n Determine the number of periods: 6 n Determine the interest rate period: 8% n Locate the value in the intersecting cell: 1. 58687 n Multiply the principal, $500, x 1. 58687 = $793. 44 n The FV of the loan is $793. 44. n Compound interest = $793. 44 - $500 = $293. 44 15 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try this example n Using Table 13 -1, find the future value and compound interest on $2, 000 invested for four years compounded semiannually at 8%. n n FV = $2, 737. 14 n CI = $737. 14 What would the simple interest be for the same loan? n $640 16 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

13. 1. 3 Find the effective interest rate n Effective interest rate is also called the annual percentage yield or APY when identifying rate of earning on an investment. n It is called APR, annual percentage rate, when identifying the rate of interest on a loan. n Effective rate: the equivalent simple interest rate that is equivalent to a compound rate 17 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Look at this example n Marcia borrowed $600 at 10% compounded semiannually. What is the effective interest rate? n Using the manual compound interest method: n Period rate interest = 10% / 2 = 5% = 0. 05 n First end-of-period principal = $600 x 1. 05 = $630 n Second end-of-principal = $630 x 1. 05 =$661. 50 n Compound interest after first year = $61. 50 18 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Effective interest rate Annual effective interest rate = $61. 50 $600 Multiplied by 100% = 0. 1025 x 100% = 10. 25% Using the table method (Table 13 -1): The table value is 1. 10250. Subtract 1. 00 and multiply by 100%. The effective rate is 10. 25% 19 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

13. 1. 4 Find the interest compounded daily using a table n Table 13 -2 gives compound interest for $100 compounded daily (using 365 days as a year. ) n Pay attention to the table value given. Table 13 -2 uses $100 as the principal amount; other tables may use $1, $10 or other amounts. n Using Table 13 -2 is exactly like using Table 11 -2 which gives the simple interest on $100. 20 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Look at this example n Find the interest on $800 at 7. 5% annually, compounded daily for 28 days. n Divide the principal by $100 as you are using Table 13 -2. [$800 ÷ 100 = 8] n Find the corresponding value by intersecting the number of days (28) and annual interest rate (7. 5%) = 0. 5769413 n Multiply this value by 8 = $4. 62 n The compounded interest is $4. 62 21 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try this example n Find the interest on $1, 000 for 30 days compounded at a 6% annual rate. n Answer: Divide $1, 000 ÷ 100 = 10 Locate the cell where 30 days and 6% intersect to determine the value: 0. 4943279 Multiply by 10. The interest is $4. 94 22 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

13. 2 Present Value n Find the present value based on annual compounding for one year. n Find the present value using a $1. 00 present value table. 23 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

13. 2. 1 Find the present value based on annual compounding for one year. n Suppose you want to go on a long vacation in a couple of years…or pay for your child’s college education. How much money would you have to invest right now to be able to pay for it? n Using the concepts of compound interest, you can determine amounts needed now to cover expenses in the future. n The amount of money you set aside now is called present value. 24 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Present value n The simplest case would be annual compounding interest for one year: the number of interest periods is 1 and the period interest rate is the annual interest rate. n Principal (present value) = future value 1 + annual interest rate* *decimal equivalent 25 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Look at this example n Find the amount of money that The 7 th Inning needs to set aside today to ensure that $10, 000 will be available to buy a new large screen plasma television in one year if the annual interest rate is 4% compounded annually. n PV = 10, 000 1. 04 n = $9, 615. 38 An investment of $9, 615. 38 at 4% would have a value of $10, 000 in one year. 26 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try these examples n Calculate the amount of money needed now to purchase a laptop computer and accessories valued at $2, 000 in a year if you invest the money at 6%. n $1, 886. 79 n John wants to replace a tool valued at $150 in a year. How much money will he have to put into a savings account that pays 3% annual interest? n $145. 63 27 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

13. 2. 2. Use a $1. 00 present value table n Using a present value table is the most efficient way to calculate the money needed now for a future expense or investment. n Table 13 -3 shows the present value of $1. 00 at different interest rates for different periods. 28 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

How to use the table 1. Find the number of interest periods: multiply the time period in years by number of interest periods per year. Interest periods = number of years x number of interest periods per year. 2. Find the interest rate: divide the annual interest rate by the number of interest periods per year. Period interest rate = annual interest rate Number of interest periods per year 29 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Using the table (continued) 3. Select the periods row corresponding to the number of interest periods. 4. Select the rate period column corresponding to the period interest rate. 5. Locate the value in the cell where the periods row intersects the rate-period column. 6. Multiply the future value by value from step 5. 30 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Look at this example n The 7 th Inning needs $35, 000 in 4 years to buy new framing equipment. How much should be invested at 4% interest compounded annually? n 4 periods at 4% shows a value of 0. 85480 n Multiply this value by $35, 000 n The result is $29, 918. n They must invest $29, 918 at 4% compounded annually for four years to have $35, 000. 31 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try these examples. n How much money would you have to invest for 5 years at 6% paid semi-annually to make a down payment of $20, 000 on a house? n n $14, 881. 80 How much money would you have to invest for 3 years at 10% paid semi-annually to purchase an automobile that costs $20, 000? n $14, 924. 40 32 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved