MA 111 Worksheet 3 2 Interest 1 Suppose we have 1000 and we want to calculate its value after a 5 increase There are two equivalent ways we have learned to do this 5 1000 100 Way 2 1000 1 05 Way 1 1000 We will be using the second way because it is simpler just a single multiplication Note the specific type of question we usually are interested in How much money will I have after some period of time We are not only interested in the increase but rather what the total amount will be 2 Annual percentage rate This is the interest rate being charged paid on a sum of money during one year s time It is often abbreviated APR For the following examples I will use an APR of R 5 which as a decimal equals 5 r 100 0 05 I will use R when expressing the rate as a percent and r when R expressing the rate as a decimal where r 100 3 Principal The principal is the amount of money that is earning interest For the following examples I will use a starting principal of 1000 4 Simple interest over one year Example 1 05 1000 1050 After one year of simple interest we have 1050 5 Simple interest over three years For simple interest we just use the APR and the starting principal to figure interest earned Example Interest from first year 0 05 1000 50 Interest from second year 0 05 1000 50 Interest from third year 0 05 1000 50 Total interest 150 Final amount 1150 which equals 1000 3 0 05 1000 or 1000 1 3 0 05 Formula If principal P is invested at simple annual interest rate R per year for t years the final amount A is given by A P 1 tr where r R 100 1 6 Compound interest over three years compounding annually For compound interest the interest that has already been earned is added to the principal That is we will earn interest on the interest we ve already accrued Example Interest from first year 0 05 1000 50 Principal after first year 1050 Interest from second year 0 05 1050 52 50 Principal after second year 1102 50 Interest from third year 0 05 1102 50 55 13 Principal after third year 1157 63 Shorter We can combine the two steps interest and revised principal into one step After the first year 1000 1 05 1050 After the second year 1050 1 05 1102 50 After the third year 1102 50 1 05 1157 63 Even shorter We can combine these three steps into one After three years 1000 1 05 1 05 1 05 1157 63 Shortest We can use an exponent to indicate the number of years that is the number of factors of 1 05 After three years 1000 1 05 3 1157 63 Formula If principal P is invested at annual interest rate R compounded every year for t years the final amount A is given by A P 1 r t where r R 100 7 Compound interest over twenty years compounding annually Example 1000 1 05 20 2653 30 8 Compound interest over one year compounding monthly We first have to find the monthly interest rate To do this divide the APR r by 12 Example After one month 1000 1 0 05 1004 17 12 12 After twelve months 1000 1 0 05 1051 16 12 9 Compound interest over t years compounding monthly Formula If principal P is invested at annual interest rate R compounded every month for t years the final amount A is given by r 12t A P 1 12 where r R 100 Example After three years 1000 1 2 0 05 36 12 1161 47 10 Compound interest over t years compounding daily Formula If principal P is invested at annual interest rate R compounded every day for t years the final amount A is given by r 365t A P 1 365 where r R 100 Example After three years 1000 1 0 05 1095 365 1161 82 11 Compound interest over t years compounding continuously We might consider compounding more often How much does this help us 05 3 8760 Example Compound hourly After three years we have 1000 1 8760 1161 833745 05 Example Compound every minute After three years we have 1000 1 525600 3 525600 1161 834234 It is possible to imagine in the limit compounding continuously all the time It turns out that a simple formula involving the number e results Formula If principal P is invested at annual interest rate R compounded continuously for t years the final amount A is given by A P ert where r R 100 On my calculator the e button is below the button Example After three years 1000 e0 15 1161 834243 12 Reviewing the above results we see there is a big jump in the final amount when we went from simple to compound interest compounding annually There is another big jump when we compound monthly instead of annually Beyond that there is not a lot to be gained from using shorter time periods for compounding In practice most banks will use monthly compounding for everything but may use daily compounding e g for outstanding credit card balances 3 13 Some Problems to Try SHOW YOUR WORK a What is the final amount if you invest 5000 for 30 years at 4 2 simple interest b What is the final amount if you invest 5000 for 30 years at 4 2 interest compounded annually c What is the final amount if you invest 5000 for 30 years at 4 2 interest compounded quarterly four times a year d What is the final amount if you invest 5000 for 30 years at 4 2 interest compounded continuously e You want to buy a 25 000 car in 5 years How much should you invest today at 5 3 interest compounded monthly to achieve that amount f Suppose you invest 1000 for one year at 4 interest compounded monthly What interest rate R compounded annually would yield the same final amount This is called the Annual Percent Yield APY g If you invest 10 000 today at 3 5 interest compounded monthly how many years will it take to double your investment 4