The Regular Payment of an Annuity with technology

UNIT 7 Annuities Date Lesson Text TOPIC Homework Dec. 7 7.1 7.1 The Amount of an Annuity with technology Pg. 415 # 1 3, 5 7, 12 **check answers withti-83 Dec. 9 7.2 7.2 The Present Value of an Annuity with technology Pg. 423 # 1 3, 5 8, 12 ** 5 needs TI-83 Dec. 12 7.3 7.3 The Regular Payment of an Annuity with technology Pg. 430 # 2 4, 6 8, 11, 16 **check answers withti-83 Dec. 13 7.4 7.6 What is a Mortgage with technology Pg. 445 # 1 5 **check answers withti-83 Dec. 14 7.5 7.6 Amortizing a Mortgage with technology QUIZ (7.1 7.3) Pg. 450 # 1 5, 8, 10, 13 **check answers withti-83 Dec. 15 7.6 Re-Cap of Annuities with technology Pg. 442 # 1 3, 5, 6, 7b, 8b OR WS 7.6 Use formulas only but check with TI-83 Dec. 16 OPT 7.7 Amortizing a Mortgage II with technology ANNUITIES TECHNOLOGY QUIZ In class with TI-83 if needed Pg. 461 # 1-4 Dec. 19 7.8 Review for Unit 7 Test Pg. 468 # 1 9, 11, 16, 17, 18 Dec. 21 7.9 TEST UNIT 7

MAP 4C Lesson 7.1 The Amount of an Annuity Ordinary Simple Annuities Ex. 1 Using a Table An annuity is a series of equal payments made at regular intervals. In an ordinary simple annuity, payments are made at the end of each compounding period. The amount of an annuity is the sum of the regular payments plus interest. Suppose $450 is deposited at the end of each quarter for 1.5 years in an investment account that earns 10% per year compounded quarterly. Complete the following table to determine the amount of the annuity? Quarter Starting Balance Interest Earned (2.5%) Deposit Ending Balance 1 $0.00 $450.00 $450.00 2 $450.00 $450.00 x 0.025 = $11.25 $450.00 $911.25 3 $911.25 $450.00 4 $450.00 5 $450.00 6 $450.00 Total There has to be a quicker way.

There is! a.k.a. Future Value (FV) The amount 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. Ex. 2 In the annuity in Ex. 1, $450 is deposited at the end of each quarter for 1.5 years at 10% per year compounded quarterly. a) Use the formula to determine the amount of the annuity? The regular payment is $450, so R = 450. 0.10 i 0.025; n = 1.5 x 4 = 6 4 Substitute R = 450, i = 0.025 and n = 6 into the amount formula. FV n ] R[( 1i) 1 i If you have a direct entry calculator, ENTER is the same as =. b) How much interest does it earn?

If using the TI-83, press [2 nd ] [x -1 ] [1] Ex. 2 In the annuity in Ex. 1, $450 is deposited at the end of each quarter for 1.5 years at 10% per year compounded quarterly. a) Use the TVM solver to determine the amount of the annuity?

b) How much interest does it earn? Annuities and regular savings Annuities are often used to save money for expenses such as a car, a down payment for a house, or a vacation. They are also used to save for education and retirement. Relatively small, regular deposits can accumulate to large sums of money over time. Ex. 4 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? You could use the formula or the TVM Solver to solve this problem. You must be able to use both. Amira Bethany N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN Ex. 4 illustrates the power of time on the value of money and the advantage of starting to save early. Pg. 415 # 1 3, 5 7, 12

MAP 4C Lesson 7.2 The Present Value of an 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. 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 = 700. 0.054 i 0.0045; n = 8 12 Substitute R = 700, i = 0.0045 and n = 8 into the present value formula. PV [ n ] R 1(1 i) i If you have a direct entry calculator,

Method 2 Use the TVM Solver N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN 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 PV [ n ] R 1(1 i) i N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN

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. 1 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.) N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN b) How much interest will she pay? Pg. 423 # 1 3, 5 8, 12

MAP 4C Lesson 7.3 The Regular Payment of an Annuity 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. 1 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? Rearranged Future Value Formula R Ai n 1i 1 Ex. 2 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

Ex. 3 Pershang borrows $9500 to buy a car. She has 2 options to repay the loan. Option A: 36 monthly payments at 6.9% per year compounded monthly Option B: 60 monthly payments at 8.4% per year compounded monthly a) Use the TVM Solver to determine her monthly payment in each case? Option A N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN Option B N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN b) How much interest will she pay for each option? Option A Option B c) Explain why Pershang might choose each option. Option A Option B Pg. 430 # 2 4, 6 8, 11, 16

MAP 4C Lesson 7.4 What is a Mortgage? What is a mortgage? A mortgage is a lien on a property/house that secures a loan and is paid in installments over a set period of time. The mortgage secures your promise that you'll repay the money you've borrowed to buy your home. What financial requirements must be met to qualify for a mortgage? Must have employment, and your mortgage payments, interest and property tax must not be more than 32% of your gross pay, and your payments, interest and property tax, heating and other monthly debt cannot exceed 40%. What is the minimum down payment required for a mortgage? Typically, a minimum down payment of 25% of the total loan balance is required to qualify for a loan without having to pay private mortgage insurance. How often is the interest compounded? Canadian law only allows mortgages to be compounded semi-annually. What is the difference between the amortization period and term of a mortgage? The amortization period is the length over which the total cost of the mortgage will be paid back. The term of a mortgage is shorter, and it is a period of time over which a certain interest rate will be paid on the mortgage. The interest rate must be renegotiated at the end of each term in order to begin the next term. What amortization periods and terms are commonly available? Typical mortgage was amortized over 25 years. Shorter amortizations do exist, such as 10, 15, and 20 years. Yet longer amortizations of 30, 35, 40 and 50 years are becoming more and more common. Mortgage terms can be from 1 year up to 25 years, typically. How often can mortgage payments be made? Monthly, semi-monthly, bi-weekly and weekly. You can also make accelerated payments. Accelerated payments are exactly half of a monthly payment amount, collected every two weeks. For example if the monthly payment is $1,000 then the bi-weekly payment will be $500. This saves you money because you pay an extra $1,000 over a twelve month period. If you pay $1,000 per month x 12 months = $12,000 in payments for the year, but if you pay bi-weekly then it is $500 X 26 = $13,000. The $1,000 a month payment is multiplied by 12, then divided by 26. This equals a bi-weekly payment of $461.54 - at the end of the year you will have paid $12,000 Conventional mortgage or a high-ratio mortgage? Conventional mortgage - If you have at least 25% of the purchase price (or appraised value if this is lower than the purchase price) as a down payment, you can apply for a conventional mortgage. Does not normally need to be insured. High-ratio mortgage - If you have between 5% and 25% of the purchase price as your down payment, you can apply for a high-ratio mortgage. Usually these have to be insured through CMHC (Canada Mortgage and Housing Corporation)

Open or closed mortgage? An open mortgage allows you to pay off part or the entire mortgage at any time without penalties. Open mortgages usually have short terms of six months or one year. The interest rates are higher than those for closed mortgages with similar terms. A closed mortgage cannot be paid off at any time without penalty. Fixed-rate or variable-rate mortgage? For a fixed rate, the interest rate is locked in for the term of the mortgage. For a variable rate mortgage, the interest rate is not locked in and will change in accordance with the change of the PRIME interest rate. It is possible the interest rate will rise over the course of the mortgage. Mortgage interest Under Canadian law, interest on mortgages can be compounded at most semi-annually. rates However, mortgage payments are often made monthly. These monthly payments form an annuity whose present value is the principal originally borrowed. Since the payment and compounding period are different, we cannot calculate the monthly payment by using the formula for the present value of a n ordinary simple annuity. We use the TVM Solver instead. To represent monthly payments and semi-annual compounding, we set P/Y = 12 and C/Y = 2. Ex. 1 The Smiths take out a mortgage of $210 000 at 5% per year compounded semi-annually for 25 years. a) What is their monthly payment? N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN ii) If they choose to make a 15% down payment, how much will they need? b) What is the total interest paid over the 25 years?

Ex. 2 Laura and Charles have to pay a 1.25% land-transfer tax on the home they bought. If the purchase price of the house was $195 000. How much is the tax? Ex. 3 Liban buys a $150 000 condo and decides to make bi-weekly payments. The bank offers a 20 year mortgage at 3.4%/a compounded semi-annually. What is his bi-weekly payment? N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN Ex. 4 Solomon buys a $210 000 cottage and makes a 20% down payment. The bank offers a 25 year mortgage at 4.1%/a compounded semi-annually. What is his monthly mortgage payment? N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN Pg. 445 # 1 5 Use TI-83 for # 5b

MAP 4C Lesson 7.5 Amortizing a Mortgage Amortizing a mortgage A mortgage is amortized when both the principal and the interest are paid off with a series of equal, regular payments. For example, in the first example, the mortgage was amortized by making monthly payments of $1221.37 over an amortization period of 25 years. To simplify the math, we assumed that the interest rate was fixed for the entire amortization period. In reality, mortgage interest rates are only fixed for a shorter length of time called the term of the mortgage. The term normally ranges from 6 months to 10 years. At the end of the term, the mortgage must be paid off or renewed at the current rate of interest. Amortization Table We can use an amortization table to analyse how a mortgage is repaid. The amortization table gives a detailed breakdown of the interest and principal paid by each payment and the loan balance after the payment. This table shows the amortization of a small loan. How much interest is paid over the term of the loan? What is the total amount paid over the period of the loan?

Ex. 2 Below is a partial amortization table for the Smith s mortgage. a) How much interest and principal is paid in the 5 th payment? b) How much do the Smiths still owe after this payment? c) What is the outstanding balance after 6 months? d) Compare the interest and principal paid in the 1 st 6 months of the mortgage with the interest and principal paid in the last 6 months of the mortgage. What do you notice? e) Why is the monthly payment increased for the 300 th payment? f) What percent of the total amount paid is interest? Pg. 450 # 1 5, 8, 10, 13

Pg. 442 # 1 3, 5, 6, 7b, 8b OR WS 7.6

MAP 4C Lesson 7.7 Amortizing a Mortgage II

Ex. Shivika buys a condo for $200000. She makes a 10% down payment and finances the rest at 4.9%/a compounded semi-annually amortized over 20 years. She has to decide whether to make monthly or bi-weekly payments. What would you advise her to do? Justify your answer. Monthly Bi-weekly Accelerated bi-weekly payments are half the monthly payment, made every two weeks (26 times per year). Pg. 461 # 1 4 USE TI-83

Annuities Study Guide

How to use the TVM solver to determine the amount of the annuity Calculating Mean, Median and Standard Deviation for a Sample with TI-83 Step 1 : Press [STAT] [ 1 ] for STAT Edit Enter the data into one of the lists Step 2: Press [STAT] Press right arrow key [ ] to highlight CALC Press [ENTER] or [1] for 1-Var Stats Enter the name of the list containing your data Press [2 nd ] [1] for L1, [2 nd ] [2] for L2... Press [ENTER] On TI-83 screen x = (Sample mean) Med =... (Sample median) x= (Sample standard deviation)

MAP 4C Lesson (OPTIONAL) Creating an Amortization Table An Amortization Table is a payment schedule that shows the amount that goes towards principle and interest of a loan, and the balance owed after n payments. Ex: Find the payment and the amortization schedule of a 30 year mortgage at $150,000 with an interest rate of 8% per year compounded semi-annually using a TI-83. Calculating the Payment: Press [APPS] and select Finance by pressing [ENTER] Select TVM_Solver by pressing [ENTER] Enter in the respective values for this mortgage N = 360, I% = 8, PV = 150,000, FV = 0, P/Y = 12, C/Y = 2 Compute the monthly payment, place the cursor to the PMT = selection and press [ALPHA] [ENTER] The payment is equal to -1087.07 N = 360 I% = 8 PV = 150000 PMT = -1087.07 FV = 0 P/Y = 12 C/Y = 2 PMT: END BEGIN Now set up the Functions Go to the Y= editor by pressing [Y=] Y1 = bal(x) To enter this press [APPS] [ENTER] [9] [X,T,θ,n] [ ) ] [ENTER] Y2 = Prn(X,X) To enter this press [APPS] [ENTER] [0] [X,T,θ,n] [, ] [X,T,θ,n] [ ) ] [ENTER] Y3 = Int(X,X) To enter this press [APPS] [ENTER] [ALPHA] [MATH] [X,T,θ,n] [, ] [X,T,θ,n] [ ) ] [ENTER] Now adjust the table set: Press [2nd] followed by [WINDOW] TblStart = 0 Tbl = 1 Indpnt: AUTO Depend: AUTO

Now view the table: [2nd] [GRAPH] The X column is the payment number Y1 represents the balance after payment X Y2, is the amount of payment X that went towards principle Y3 is the amount of payment X that went towards interest. To view Y3, scroll to the right. Pg. 460 # 4-6