Getting Started Pg. 450 # 1, 2, 4a, 5ace, 6, (7 9)doso. Investigating Interest and Rates of Change Pg. 459 # 1 4, 6-10

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1 UNIT 8 FINANCIAL APPLICATIONS Date Lesson Text TOPIC Homework May Opt Getting Started Pg. 450 # 1, 2, 4a, 5ace, 6, (7 9)doso May Investigating Interest and Rates of Change Pg. 459 # 1 4, 6-10 May Compound Interest: Determining Future Value Pg. 468 # 1 9, 11, 12, 15 May Compound Interest: Determining Present Value Pg. 476 # 2, 5 14 May Compound Interest: Solving Financial Problems (Use formula and check with TI-83) Pg. 486 # 1 10, 13 June Mid-Chapter Review Pg. 491 # 1, 2, 5 11, (12 14 with TI-83) June Regular Annuities: Determining Future Value QUIZ ( ) Pg. 498 # 2 11, 13, 14 June Regular Annuities: Determining Present Value Pg. 506 # 2-11 June Savings Plans and Loans Pg. 518 # 2, 4, 6-10 June Review for Unit 8 Test Pg. 522 # 1, 3a, 4 6, 8, 9a, 10, 11 15, 17 June UNIT 8 TEST June 8 & Exam Review Exam Review WS Textbook Cumulative Reviews Pg. 186 # 1 20 Pg. 382 # 1-32 Pg. 526 # 1-12, 14, 15 June 19 Take up Practice Exam/Other Questions June 23 9:30 in Room 103

2 MCF 3M Lesson 8.1 Investigating Interest simple (regular) interest - interest earned or paid only on the original sum of money invested or borrowed compound interest - interest calculated at regular periods and added to the principal for the next period ie: interest is calculated on the principal and interest already earned principal - a sum of money that is borrowed or invested term - the duration of an investment, or the time required to pay off a loan Simple interest is calculated by applying the interest rate only to the original principal amount, resulting in linear growth. Compound interest is calculated by applying the interest rate to the original principal and any accumulated interest, resulting in exponential growth. The interest rate is converted to decimal form prior to calculating interest earned. Simple interest can be calculated with the formula I = Prt, where I is the interest, earned in dollars P is the principal invested or borrowed, in dollars r is the annual interest rate, expressed as a decimal t is the time, in years The amount that a simple-interest investment or loan is worth can be calculated with the formula A = P + Prt or its factored form A = P(1 + rt), where A is the final amount of the investment or loan, in dollars P is the principal, in dollars r is the annual interest rate, expressed as a decimal t is the time, in years Tables are useful as tools for organizing calculations involving compound interest. Ex. 1 Kevin invested $4500 in a two-year regular-interest CSB that earns 7.5% simple interest annually. a) How much interest did he earn? b) How much will his investment be worth at the end of the term? amount - the sum of the original principal and the interest; given by A = P + I, where A is the amount, P is the principal, and I is the interest

3 Ex. 2 Keila had a credit card balance of $550 that was 31 days overdue. The annual interest rate on the card is 23.9%. a) How much interest did Keila have to pay? b) Explain why paying interest on an outstanding credit card balance is sometimes referred to as the cost of borrowing money. Ex. 3 Five years ago, Jason lent Matt money. Matt repaid Jason a total of $2100, which included simple interest charged at 10%. How much did Jason originally lend Matt?

4 Guaranteed Investment Certificate (GIC) - an investment purchased from a bank, trust company, or credit union, which guarantees that a specified interest rate will be the same over a fixed period. When an investment or a loan reaches the end of its term, it is said to have matured or to have reached maturity. Ex. 4 Mohsin bought a $500 Guaranteed Investment Certificate (GIC). It has a 3-year term and earns 3.25% compounded annually. How much interest will the GIC have earned at maturity? Year Principal (Amount at beginning of year) ($) Interest Earned ($) Amount at the end of the year ($) WHAT IF THE GIC HAD A TWENTY YEAR TERM? THERE MUST BE A FASTER WAY! Pg. 459 # 1 4, 6-10

5 MCF 3M Lesson 8.2 Compound Interest and Future Value Liban deposits $100 in his savings account that pays 10% per annum compounded annually. How much will he have at the end of 5 years if he does not withdraw or deposit anything in that time? future value - the final amount (principal plus interest) of an investment or loan when it matures at the end of the investment or loan period The formula for calculating the amount of an investment earning compound interest is, A = P(1 + i) n where A is the amount or future value, in dollars P is the principal, in dollars i is the interest rate per compounding period n is the number of compounding periods compounding period - each period over which compound interest is earned or charged in an investment or loan The compounding frequency determines the number of compounding periods per year. The compounding period changes the total number of periods, n, over which the interest is compounded during the term of the investment. Changing the compounding period changes the interest, i, because the annual interest rate must be adjusted to the rate that would be used for each compounding period. Ex. 1 Jayesh has $2000 to invest in a compound-interest account in which he would like to leave the money for 3 years. He considers three different compounding periods: Option A: 8%/a compounded annually Option B: 8%/a compounded semi-annually Option C: 8%/a compounded quarterly a) For each option, what is the amount at the end of 3 years? What is the interest earned?

6 b) How does changing the compounding period affect the amount of the investment? Why? Ex. 2 Alwynn invests $500 in an account that earns 6%/a compounded monthly. Peter invests $500 at the same time, but in an account that earns 6%/a simple interest. Determine the difference between their investments at the end of the 5th year. Pg. 468 # 1 9, 11, 12, 15

7 MCF 3M Lesson 8.3 Compound Interest and Present Value The formula for calculating future value can be rearranged to give the present value of an investment earning compound interest. The rearranged A n formula is: P or P A( 1 i) n ( 1 i) A is the amount or future value, in dollars P is the principal or Present Value, in dollars i is the interest rate per compounding period n is the number of compounding periods Ex. 1 When Hua was born, her parents decided to invest some money so that she could have a gift of $ on her 17th birthday to use for her post secondary education. They decided on a compound-interest government bond that paid 10% interest per year, compounded monthly. After the initial amount was invested, there would be no further transactions until the bond reached maturity. How much money must be invested today to guarantee Hua s future amount of $50 000? Ex. 2 An investment earns 7.75%/a compounded semi-annually. Determine the present value if the investment is worth $800 five years from now.

8 Ex. 3 Tony has $3000 in his savings account. He intends to buy an HD TV and invest the remainder for 2 years, compounding quarterly at an annual interest rate of 5%. He wants to have $2000 in his account 2 years from now. How much can he spend on the TV? Ex. 4 Khalil borrowed some money from his friend Humphrey. After 2 years he pays Humphrey $ to cover the loan. Humphrey charged 5%/a compounded annually. How much did Khalil borrow? Pg. 476 # 2, 5 14

9 MCF 3M Lesson 8.4 Solving Financial Problems with the TVM Solver If using the TI-83, press [2 nd ] [x -1 ] [1] When entering the values for present value or future value in the TVM Solver,consider whether the money is paid out (cash outflow) or received (cash inflow). Money paid out, such as a loan repayment or the principal of an investment, is negative. Money received, such as the final amount of an investment, is positive. Enter the values of all of the program variables except for the one you want to calculate. The remaining variable has a value of 0 because its value is unknown.the actual value of the remaining variable is then calculated and displayed.

10 Ex. 1 Find the future value or amount of $7500 invested for nine years at 8%/a, compounded monthly. How much interest was earned? Ex. 2 Maeve would like to have $3500 at the end of five years, so she can visit Europe. How much money should she deposit now in a savings account that pays 9%/a, compounded quarterly, to finance her trip? Ex. 3 What annual interest rate was charged if an $800 credit card bill grew to $ in 6 months and interest was compounded monthly? (Use formula and check with TI-83) Pg. 486 # 1 10, 13

11 MCF 3M Lesson 8.5 Mid-Chapter Review Pg. 491 # 1, 2, 5 11, (12 14 with TI-83)

12 MCF 3M Lesson 8.6 Future Value of a Regular Annuity Annuity - a series of equal deposits or payments made at regular intervals; a simple annuity is an annuity in which the payments coincide with the compounding period, or conversion period; an ordinary annuity is an annuity in which the payments are made at the end of each interval; unless otherwise stated, each annuity in this unit is a simple, ordinary annuity When Jessica turned 13 years old, her grandmother gave her $500. Jessica deposited the money into a savings account that paid 4.5%/a compounded annually. Since then, Jessica s grandmother has made an automatic deposit of $500 into Jessica s account on her birthday each year. Each payment earns a different amount of interest because it compounds for a different amount of time. The timeline shows the deposits of the annuity made into Jessica s account. For how many compounding periods has the $500 deposit on her 17th birthday earned interest? For how many compounding periods has each of the 16th, 15th, 14th, and 13th birthday deposits earned interest? To get the total (future) value of the annuity we could evaluate each deposit s value and add them up, use the TVM-Solver or the Future Value of an Annuity Formula. Future Value of an Annuity FV n R[( 1 i) 1] i where, FV (or A) is the amount, or future value, in dollars R is the regular deposit, or payment, in dollars i is the interest rate per compounding period, expressed as a decimal n is the total number of deposits

13 Ex. 1 Calculate the accumulated amount of: a) $100 invested at the end of each year for 30 years at 8% compounded annually. b) $20 invested at the end of each month for 25 years at 6% compounded monthly. Ex. 2 Jane wants to have $3000 in one year to take a trip to Spain. She has found an investment account at her bank that will pay 7.2% per year compounded monthly. What monthly payments must she make to reach her goal?

14 Ex. 3 Amira and Bethany are twins. They save for retirement as shown below. Starting at age 25, Amira deposits $1000 at the end of each year for 40 years. Starting at age 40, Bethany deposits $2000 at the end of each year for 25 years. Suppose that each annuity earns 8% per year compounded annually. Who will have the greater amount at retirement? Amira Bethany Ex. 4 Habeeba wants to save $3000 for the Japan trip in 3 years. What regular deposit should she make at the end of every 6 months in an account that earns 6% per year compounded semi-annually? You must be able to use both the TVM-Solver and the formula. Pg. 498 # 2 11, 13, 14

15 MCF 3M Lesson 8.7 Present Value of a Regular Annuity Present Value of an Annuity The present value of an annuity is the principal that must be invested today to provide the regular payments of an annuity. n is the number of payments or withdrawals The present value formula can only be used when: The payment interval is the same as the compounding period. A payment is made at the end of each compounding period. The first payment is made at end of the first compounding period. Providing for an Annuity Ex. 1 Hudson wants to withdraw $700 at the end of each month for 8 months, starting 1 month from now. His bank account earns 5.4% per year compounded monthly. How much must he deposit in his bank account today to pay for the withdrawals? Method 1 Use the PV formula The regular payment is $700, so R = i ; n = 8 12 Substitute R = 700, i = and n = 8 into the present value formula. If you have a direct entry calculator,

16 Method 2 Use the TVM Solver Ex. 2 Andrea plans to retire at age 55. She would like to have enough money saved in her account so she can withdraw $7500 every 3 months for 30 years, starting 3 months after she retires. How much must she deposit at retirement at 9% per year compounded quarterly to provide for the annuity? The amount she must deposit is the present value of the annuity. Method 1: Formula Method 2: TVM Solver

17 Repaying Loans Most loans are repaid by making equal monthly payments over a fixed period of time. These payments form an annuity whose present value is the principal borrowed. When all of the payments are made, both the principal borrowed and the interest due will have been paid. Ex. 3 Allison plans to buy a car. She can afford monthly payments of $300. The car dealer offers her a loan at 6.9% per year compounded monthly, for 4 years. The first payment will be made 1 month from when she buys the car. a) How much can she afford to borrow? The amount she can borrow is the present value of the loan. (You could use the formula or the TVM Solver.) b) How much interest will she pay?

18 When we know the amount or the present value of annuity, we can solve for the regular payment. To do this, we rearrange the appropriate formula to solve for R. We can do this before substituting for all the known values or after substituting. Ex. 4 David borrows $1200 from an electronics store to buy a computer. He will repay the loan in equal monthly payments over 3 years, starting 1 month from now. He is charged 12.5% per year compounded monthly. How much is his monthly payment? Rearranged Present Value Formula R PV i 1 1i n Pg. 506 # 2-11

19 MCF 3M Lesson 8.8 Savings Plans and Loans amortization - the process of gradually reducing a debt through instalment payments of principal and interest down payment - the partial amount of a purchase paid at the time of purchase Ex. 1 Compare the amounts at age 65 that an RRSP at 6%/a compounded annually would earn under each option. Option 1: making an annual deposit of $1000 starting at age 20 Option 2: making an annual deposit of $3000 starting at age 50 What is the total of the deposits in each situation? Option 1 Option 2

20 Ex. 2 Show how changing the payment frequency from semi-annual to weekly affects the amount of interest paid and the length of time needed to repay a loan of $ at 11%/a. For example, Joe makes semi-annual payments of $520 and interest is charged semi-annually, while Sarit makes weekly payments of $20 and interest is charged weekly. JOE SARIT

21 Ex. 3 David has a credit card balance of $ that is charged 23%/a interest, compounded monthly. He decides not to use his credit card again and to make monthly payments to pay off his debt. a) How long does it take to reduce his credit card balance of $ to 0 if he pays $1000 a month? b) If he increases his monthly payment to $1600, how much sooner is the debt paid off? c) How much will he save if he makes $1600 payments? Pg. 518 # 2, 4, 6-10