SECTION 6.1: Simple and Compound Interest

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1 1 SECTION 6.1: Simple and Compound Interest Chapter 6 focuses on and various financial applications of interest. GOAL: Understand and apply different types of interest. Simple Interest If a sum of money P, called the or is invested for t years at an annual simple interest rate of r (given as a decimal), then the interest I at the end of t years is given by I = Prt. The or A at the end of t years is A = P + I = P + Prt = P(1 + rt) Example 1: Finding Interest and Accumulated Amount Greg borrowed $1000 on a credit card that charges simple interest at an annual rate of 18%. a) What is Greg s interest for the first month? b) How much does Greg owe the credit card company after the first month?

2 2 Example 2: Finding Principal A bank deposit paying simple interest at the rate of 6% per year grew to a sum of $2500 in 8 months. What was the principal? Example 3: Finding Simple Interest Rate A person deposits $1000 into an account earning simple interest. What simple interest rate is being obtained if the amount A at the end of 8 months is $1060? Example 4: Finding the Time Suppose you deposit $100 into a bank account with a simple interest rate of 7.5%. You plan on pulling out the money when your bank account reaches $600. How long will you have to wait? RECAP on Simple Interest Whenever your problem is dealing with simple interest, you will want to use the formulas: where I = P = r = t = A = I = Prt and A = P(1 + rt)

3 3 Example 5: Motivating Compounded Interest Compounded interest is when your interest money starts making its own money. For example, compare the two situations: a) You invest $100 in a bank account with a 10% simple annual interest rate. What is your accumulated amount after 1 year? b) Same situation as above, except this time your bank decides to award you your interest every 6 months. That is, after the first 6 months your bank account is now: A = P 1 + rt = !! = $105. If you continue to invest for the full year, then you can calculate how much money you have at the end of the year by the following: A = P 1 + rt = = $ Compound Interest If a principal P earns interest at an annual rate r (given as a decimal) and interest is compounded m times per year, then the amount of money A after t years is: A = P 1 + r m!" While this is a lovely formula to use, there is an even more lovely calculator app called TVM Solver that can solve compound interest problems for us. To access this solver, do the following on your calculator: Click APPS then select 1:Finance then select 1:TVM Solver. N = mt (number of compounds) I% = Interest rate as a percentage PV = Present value (think principal) PMT = Payment amount (this is for sections 6.2 and 6.3 for now leave as 0) FV = Future value P/Y = C/Y = m (compounds per year) To solve for a value, move the cursor to that row. Then hit ALPHA and ENTER

4 4 Example 6: Finding Future Value with Compounded Interest Given an initial deposit of $1,000 and a rate of 10%, find the value after 40 years if the interest is compounded a) Annually b) Semi-annually c) Quarterly d) Monthly e) Weekly f) Daily g) Continuously Example 7: Finding Present Value with Compounded Interest Find the present value of $40,000 due in 4 years at the interest rate of 11% per year compounded daily. N = I% = PV = PMT = FV = P/Y = C/Y =

5 5 Example 8: Trust Fund A young man is the beneficiary of a trust fund established for him 16 years ago at his birth. If the original amount placed in the trust was $20,000, how much will he receive if the money has earned interest at the rate of 9%/year compounded quarterly? N = I% = PV = PMT = FV = P/Y = C/Y = Example 9: Retirement Six and a half years ago, Chris invested $10,000 in a retirement fund that grew at the rate of 11.14%/year compounded quarterly. What is his account worth today? N = I% = PV = PMT = FV = P/Y = C/Y = Example 10: Motivating Effective Rate a) Suppose you invest $1,000 in a bank account with 8% interest compounded monthly for one year. How much money do you have at the end of the year? b) Suppose you invest $1,000 in a different bank account with 8.3% interest compound yearly for one year. How much money do you have at the end of the year? c) Was one bank account more profitable than the other?

6 6 Definition: Effective Rate The r eff of an interest rate r compounded m times per year is such that r eff compounded once a year earns the same interest as the rate r compounded m times. The rate r is referred to as the nominal rate. You can compute the effective rate by going to the Finance app in your calculator: Click APPS then select 1:Finance, and then select C: Eff(. You should see : Eff( in your main screen. To find the effective rate of an interest rate r compounded m times, enter the following: Eff(annual interest rate %, # compounds per year) Example 11: Better Rates Bank A is offering a savings account at 6.6% interest that compounds quarterly, while bank B offers an account at 6.5% interest compounded daily. Which bank is offering the better deal? Example 12: Effective Rates Find the effective rate of interest corresponding to a nominal rate of 10% per year when a) compounded annually b) compounded semiannually c) compounded quarterly d) compounded monthly