8.1 Simple Interest Interest: The money earned from an investment you have or the cost of borrowing money from a lender. Simple Interest: "I" Interest earned or paid that is calculated based only on the original amount invested or borrowed. Principal: "P" The sum of money that is borrowed or invested at the beginning of the loan term. Also called the present value "PV". Amount: "A" The total value of an investment or loan that has been earned, or that is owed at the end of the loan term. also called the future value "FV". The amount at the end of a loan is equal to the principal earned or borrowed + the accrued interest. A = P + I Simple Interest Formula: Interest I = Prt Principal time (YEARS) interest rate Ex.1 Julie borrows $15 000 at 6.8%/a simple interest. How much will she owe if she repays the loan at the end of 2 years? What would she owe if she repaid the money at the end of 10 years? Ex.2 Phil borrows $540 for 75 days by taking a cash advance on his credit card. The interest rate is 21.7%/a simple interest. How much will Phil need to pay back to his credit card company? 8.1 Assignment: p. 481 #3, 4, 5bdef, 6-8
8.2 Compound Interest Formula: where P is the principal or starting amount A is the amount at the end of the term i is the interest rate per compounding period n is the total number of compounding periods For example if money is invested in an account for 7 years and earns 10%/a compounded semiannually. Then after 6 months you receive the first 5% and at the end of 12 months you receive the next 5% for a total of 10% in that year. So i = 10%/2 = 5% Each year there are two compounding periods and the money is invested for 7 years. So n = 2 x 7 = 14 Ex. 1: Determine the amount of $7 500 at 3.6%/a compounded quarterly for 2 years. How much interest was earned? Ex. 2: Complete the table below Rate of Compound Interest Per Year Compounding Period Time Interest Rate per Compounding Period, i Number of Compounding Periods 5.4% semi annually every 6 months 5 5.4% / 2 = 2.7% =0.027 n = 2 x 5 = 10 3.6% monthly 3 2.9% quarterly 7 2.6% weekly 10 months 8.2 Assignment: p.490 # 4, 6, 7, 9-11, 17
8.3 Compound Interest: Present Value When finding the Present Value P, we are using the same formula. However, this time we know the amount A at the end of the loan or investment. Compound Interest Formula: Isolate P: Ex.1: How much must be invested now in order to have $15 000 in 10 years if the money earns 4.4%/a compounded semi-annually. Ex. 2: Jim invested $5000 that he would like to grow to be at least $50 000 by the time he retires in 25 years. At what annual interest rate, compounded monthly, will Jim need to invest the money? 8.3 Assignment: p. 498 #3-6, 8-11
8.4 Annuities: Future Value An annuity is a series of regular, equal payments paid into, or out of, an account. When deposits are made at the end of each equal time interval, the annuity is called an ordinary annuity. The total value of all the deposits at the end of the last time interval is called the amount of the annuity. The compounding period is the time between deposits. now Cassandra spends $ every month at Starbucks. Her teacher suggests that she should save the $ instead. She decides to take her teacher's advice and invests the $ every month into a bank account that pays 6%/a compounded monthly. How much money will Cassandra have at the end of 2 years? end of week 1 2 3 4 23 24 (1.005) 23 (1.005) 22 (1.005) 1 So we want the sum of: + (1.005) 1 + + (1.005) 22 + (1.005) 23 Notice it's a geometric sequence!
Amount of an Ordinary Annuity : A deposit R is made at the end of each period for n periods. The deposits earn interest at a rate of i per compounding period. The amount of the deposits at the end of the nth period A is the amount or future value of the annuity R is the regular deposit i is the interest rate per compounding period, expressed as a decimal n is the number of compounding periods. Ex1. Tom wants to make monthly deposits to have $75000 in his account at the end of 15 years. Interest is 4.8%/a compounded monthly. a. How much should he deposit each month? b. How much interest did Tom earn? 8.4 Assignment: p.511 #1-4, 5bd, 6, 7, 9.
8.5 Annuities: Present Value Present Value of an Ordinary Annuity The amount of money needed to finance a series of regular withdrawals. A payment, R, is withdrawn from an account at the end of each period for n periods. The account earns interest at a rate of i per period. The amount required in the account at the beginning of the first period is: Cassandra would like to focus on her studies and quits her job. She would still like to spend $ every month at Starbucks for 1 year. How much money does she need in her bank account to withdraw $ each month if interest is 6%/a compounded monthly? end of week now 1 2 3 4 24? (1.005) 1 (1.005) 2 (1.005) 3 23 (1.005) 23 So we want the sum of: Notice it's a geometric sequence! (1.005) 1 (1.005) 2 (1.005) 3 (1.005) 23 + + + +
Ex1. Matteo borrowed $7500, at 3.8%/a compounded monthly, to purchase a car. He wants to pay off his car in 19 months. a. How money does he need to pay each month? b. How much will he pay for his car? Assignment 8.5: p. 520 # 3bc, 4-8.