Simple Interest, Future Value, Present Value, and Effective Rate

Math 1313 Chapter 5 – Section 5.1 Simple Interest, Future Value, Present Value, and Effective Rate Simple Interest is interest that is computed on the original principal only. Formula: I = Prt, where P is the principal, r is the interest rate and t is time (in years). Accumulated Amount is the sum of the principal and interest after t years. Formula: A = P(1 + rt) P, r and t have the same meaning as above. Example 1: Find the simple interest on a $1,350 investment made for 2 years at an interest rate of 4% per year. Example 2: Find the accumulated amount at the end of 7 months on a $900 bank deposit paying simple interest at a rate of 5% per year. Compound Interest is earned interest that is periodically added to the principal and thereafter itself earns interest at the same rate. Future Value with Compound Interest Formula: A = P (1 + i ) n , where i = r and n = mt. m A stands for the Future Value or the accumulated amount at the end of n conversion periods. A conversion period refers to the interval of time between successive interest calculations. P stands for the Present Value or principal. r stands for the interest rate per year. m stands for the number of conversion periods per year. t stands for time (in years). Example 3: Find the future value of $2,900 invested at 6.25% per year compounded monthly for 4 years. Recall: A = P(1 + i ) n and that P stands for present value. Why would we want to find P? Well in certain instances an investor may wish to determine how much money he/she should invest now, at a fixed rate of interest, so that he/she will realize a certain sum of money at some future date. So, solving the Future Value Formula for P we obtain the Present Value with Compound Interest Formula: P = A(1 + i ) − n , where A, i and n have the same meaning as before. Example 4: Find the present value of $5,500 due in 3 years at an interest rate of 2.5% per year compounded semiannually. Example 5: Tamara would like to take a vacation to the Caribbean Islands in 2 years. She invests $1,500 in a savings account that pays 5% per year compounded semiannually. How much will she have available for her vacation in 2 years? Example 6: Charlie recently found out that he is going to be a grandfather. He’s decided to plan ahead and invest some money in an account for his new grandchild’s college education in 18 years. He’s invested $5,000 in an account that pays 6% per year compounded quarterly. He plans to leave this investment in this account for 18 years. How much money will his grandchild have towards his/her college education in 18 years? Example 7: Tyrone invested a sum of money 5 years ago in an account that paid 4.75% per year compounded quarterly. He recently closed the account and received $11,671.00. How much did he originally invest in this account? Example 8: Kaylin is planning on buying a home in 6 years. She’d like to have $6,000 for a down payment in 6 years. Her credit union has an account that will pay 3% per year compounded monthly. How much must she invest now in this account to have the desired funds available in 6 years? Effective Rate m Effective Rate of Interest Formula: r⎞ ⎛ reff = ⎜1 + ⎟ − 1 ⎝ m⎠ where reff is the effective rate of interest, r is the nominal interest rate per year, and m is the number of conversion periods per year Note: The effective rate of interest formula shows that money invested at simple interest earns the same amount of interest in one year as money invested at r% per year compounded m times a year. Example 9: Find the effective rate corresponding to the nominal rate of 10% per year compounded monthly.