Activity 3.1 Annuities & Installment Payments


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1 Activity 3.1 Annuities & Installment Payments A Tale of Twins Amy and Amanda are identical twins at least in their external appearance. They have very different investment plans to provide for their retirement. Study both plans and then decide who you think will have more money in her RRSP when they both reach 65 years of age if their investments yield 9% per annum compounded annually. Amy Amanda Investing is all math. Harry Markowitz Nobel Prize for Economics, 1980 RRSP = Registered Retirement Savings Plan This is an account into which money can be placed so that it can compound freely without tax until withdrawn during retirement. I plan to put $1 000 each year into an RRSP account beginning on my birthday when I turn 20 years of age. When I turn 29, I will make my final contribution of $ Then I will leave the money invested in my RRSP and let it grow, but I will make no further contributions. Q How much will Amy contribute to her RRSP between ages 20 and 30? I want to have some spending money while I m young, so I will not open an RRSP until I reach 30 years of age. However, when I turn 30 years of age, I will put $1 000 into an RRSP account each and every year until I turn 64. Then I will make my final contribution of $ Q How much will Amanda contribute to her RRSP between ages 30 and 65? Both Amy and Amanda plan to make deposits of $1 000 into their RRSP accounts every year. Any such series of equal payments made at regular intervals of time is called an annuity. Annuities can be used to build assets or to pay off debt as in installment buying or the discharge of a mortgage. 56 PART 2: ACTIVITY 3.1 ANNUITIES & INSTALLMENT PAYMENTS
2 To calculate the value of Amy s RRSP at age 65, we cannot simply add her $1 000 contributions together, because the contributions are made at different times and therefore have grown into different amounts. To determine the value of Amy s RRSP at age 65, we must compute the accumulated value of each contribution on Amy s 65th birthday and then add them together, as shown in the time diagram. Amy s Plan To find the accumulated value of Amy s RRSP when she reaches age 65, we sum the accumulated values of all the contributions shown in the column under Age 65, using the fact that it is a geometric series of 10 terms with first term 1 000(1.09) 36 and common ratio (1.09). Accumulated value ($) = 1 000(1.09) (1.09) (1.09) (1.09) 45 = 1 000[(1.09) 46 (1.09) 36 ] / 0.09 = That is, Amy s RRSP will have an accumulated value of $ on her 65 th birthday. Amanda s Plan To find the accumulated value of Amanda s RRSP when she reaches age 65, we sum the accumulated values of all the contributions, using the fact that it is a geometric series of 35 terms with first term 1 000(1.09) and common ratio (1.09). Accumulated value ($) = 1 000(1.09) (1.09) (1.09) (1.09) 35 = 1 000[(1.09) 36 (1.09)] / 0.09 = That is, Amanda s RRSP will have an accumulated value of $ on her 65 th birthday. Compare the contributions made by Amy and Amanda and the values of their RRSPs at age 65. Are you surprised? Explain why or why not. PART 2: ACTIVITY 3.1 ANNUITIES & INSTALLMENT PAYMENTS 57
3 Activity 3.2 Buying on the Installment Plan Using a Graphing Calculator to Verify your Answer We can verify our answer by using the TVM Solver (timevalueofmoney) on the FINANCE menu of the TI83 Plus. To activate this, we press APPS ENTER ENTER. Then we enter the number of payments N, the interest rate I, the principal PV, the future value FV, the number of payments per year P/Y, and the number of compoundings per year C/Y, as in the display. We place the cursor opposite PMT(payment) and press ALPHA [SOLVE]. The display shows verifying that equal payments of $ are required. N = 60 I% = 12 PV = PMT = 0 FV = 0 P/Y = 12 C/Y = 12 PMT: END BEGIN Worked Example 1 Most people purchase cars on the installment plan. The lending institution advances a loan of $P called the principal. The borrower repays the loan in equal installments over a given number of years, n (called the term of the loan). Both parties agree to a particular interest rate i and compounding period (usually a month), and the lending institution informs the borrower what equal installment payments x are required to repay the principal. This example shows how to calculate the value of x. A car loan of $ is offered at 12% per annum compounded monthly 1. What equal monthly payments are required to repay the loan over a term of 5 years? Solution To determine the amount of the monthly payments, we compute the present value of all the 60 payments made over the 5year term. The sum of these present values should equal the amount of the loan, i.e., $ Observe that when we calculate future value of money, we multiply by the growth factor, but when we calculate present value of future payments, we divide by the growth factor. The total present value of all the payments is given by: x (1.01) + x (1.01) +... x + 2 (1.01) + x 59 (1.01) 60 x This is a geometric series with first term (1.01) and common ratio (1.01) 1. The sum of this series is 100x [1 (1.01) 60 ]. Since the total present value of all the payments should be equal to the principal, i.e., $20 000, we write 100x [1 (1.01) 60 ] = Solving this equation for x yields x = That is, 60 monthly payments of $ will repay the loan in 5 years. N = 60 I% = 12 PV = PMT = Note: 12% per annum compounded monthly means the interest is 1% per month and is calculated at the end of each month. 58 PART 2: ACTIVITY 3.2 BUYING ON THE INSTALLMENT PLAN
4 Activity 3.3 Calculating the Outstanding on a Spreadsheet In Worked Example 1, we showed how to calculate the payment amounts in an annuity. However, when we have an installment loan, we already know the payment amount, but often wish to determine the balance owing at a particular time. The following worked example shows how we can use a spreadsheet to display the balance after each payment. Worked Example 2 For the $ car loan in Worked Example 1: a) What is the balance of the loan after the 9th payment? b) How much of the 9th payment is allocated to interest, and how much to the principal? Solution To display on a spreadsheet the monthly balances of the $ car loan described on the previous page, we enter the headings and formulas shown in the spreadsheet below. A B C D E 1 Payment # Initial Interest on Payment Final =$B3* =$B3+C3 D3 4 =$A3+1 =$E3 =$B4* =$B4+C4 D4 5 =$A4+1 =$E4 =$B5* =$B5+C5 D5 We use the Fill Down command to extend these formulas to the 11 th row of the spreadsheet (i.e., payment #9). When we display the numbers (rather than the formulas), we obtain the display below. A B C D E 1 Payment # Initial Interest on Payment Final The 11 th row reveals that the balance before the 9 th payment is $ , but after the 9 th payment the (final) outstanding balance is $ The difference, $ $ or $265.18, is the amount that has been allocated to the principal. Since the total payment is $444.89, then the interest included in that payment is $ $ or $ after the 9 th payment To verify that we have entered appropriate formulas into the spreadsheet, we can use the Fill Down command to extend the formulas to the 62 nd row (60 th payment). We then display the numbers and check that the balance after the 60 th payment is zero (or very close to it). PART 2: ACTIVITY 3.3 CALCULATING THE OUTSTANDING BALANCE ON A SPREADSHEET 59
5 Activity 3.4 Exploratory Activity Buy or Lease a Car Which Is the Cheaper Option? Many people debate whether it s cheaper to buy or to lease a car. When you buy a car, you negotiate with the dealer on a purchase price. Then you pay either the purchase price (plus tax, money levied by the government on income and sales) or a portion of the purchase price, called the down payment, and finance the rest. The part that you finance is the principal of an installment loan that is paid off in equal payments over the term of the loan (usually 3 to 5 years). When you lease a car, the dealer purchases the car on your behalf. The purchase price that you negotiate with the dealer is called the cap cost (short for capitalized cost). The lease agreement entitles you to the use of the car for a fixed term (usually for 2 or 3 years) and the option to buy the car at the end of the term at a preset price called the residual (value). In return for the use of the car during the term of the lease, you make monthly payments that are substantially less than if you were buying the car, but then you do not own the car at the end of the lease unless you exercise your option to buy. The mathematics of each financial arrangement are shown below. The Finances of Buying a Car 1 Reach agreement with the dealer on the purchase price of the car including optional extras. 2 Add the appropriate GST and PST to the purchase price. 3 Subtract the down payment from 2. 4 Agree upon an interest rate and term for an installment loan with principal obtained in 3. 5 Calculate the monthly payments required to pay off the principal of the loan in 4. The Finances of Leasing a Car 1 Reach agreement with the dealer on the purchase price of the car including optional extras. This is the cap cost. 2 Add to the cap cost the acquisition fees that the dealer charges for administering the lease. 3 Subtract the down payment from 2 to obtain the adjusted cap cost. 4 Subtract the residual from the adjusted cap cost to obtain the amount to be financed. 5 Agree to the terms of the payment schedule based on the amount obtained in 4. 6 Calculate the monthly payments required to pay off the principal of the loan in 5, and add the GST and applicable PST to each payment. 60 PART 2: ACTIVITY 3.4 EXPLORATORY ACTIVITY
6 Suppose you are interested in a car that sells for $ You have saved a down payment of $6 000 that you want to put into a lease or purchase. A car dealer provides you with the purchase agreement and the lease agreement shown below. Read the terms of each offer and fill in the missing data. The Purchase Agreement Purchase Price: $ Loan Information: 9% annually compounded monthly term of 3 years 1 The purchase price plus tax is: $ 2 The amount to be financed is: $ 3 The monthly payment is: $ 4 The accumulated value of all the monthly payments at the end of 3 years, assuming that they could have been invested at 3% per annum compounded monthly, is: The Lease Agreement Cap Cost: $ Acquisition Fees: $5 286 Residual: $ Lease Information: 9% annually compounded monthly term of 3 years 5 The cap cost is: $ 6 The adjusted cap cost is: $ 7 The amount to be financed is: $ 8 The monthly payment (including GST and PST) is: $ 9 The accumulated value of all the monthly payments at the end of 3 years, assuming that they could have been invested at 3% per annum compounded monthly, is: $ $ 10 Write a brief report comparing the costs of purchasing and leasing for the example given here. Discuss the advantages of each form of financing a car and describe the kinds of car use that would favour each method. In this activity, we have focussed on price only. Some people argue that leasing has the advantage that it doesn t tie up capital that could be invested elsewhere at higher rates of return. Tax considerations might also affect your decision whether to lease or buy. PART 2: ACTIVITY 3.4 EXPLORATORY ACTIVITY 61
7 Activity 3.5 Exercises 1 Write a sentence to explain what is meant by each of the following terms in an auto lease: a) cap cost c) acquisition fees e) residual value b) adjusted cap cost d) term 2 List all the parameters (i.e., variables) that are necessary to define an annuity. 3 Is smoking really expensive? The average smoker spends about $1 700 per year on cigarettes. If this amount were deposited at the end of each year in an account earning 6% compounded annually, how much would be accumulated at the end of 20 years? 4 What monthly payments earning interest at 6.75% per annum compounded monthly will accumulate to $ at the end of 25 years? 5 Calculate the monthly payments required for a $ mortgage to be paid off over 30 years at 9.0% per annum compounded monthly. 6 How long would it take to accumulate: a) $ with deposits of $2 000 per year earning 8% per annum compounded annually? b)$ with deposits of $5 000 per year earning 6% per annum compounded annually? c) $ with deposits of $5 000 per year earning 12% per annum compounded annually? 7 A financial institution offers consumers a choice of two different 10year savings plans. On a fixed amount of money deposited at the beginning of every year for 10 years, Plan A pays 5% per annum interest (compounded annually) in each of the first five years and 4% per annum interest in each of the remaining five years. Plan B offers 4% in the first five years and 5% in the remaining five years. Which plan, if any, offers the higher return? 8 a) Calculate the monthly payment on a mortgage of $180 7% interest per annum compounded monthly and amortized over 30 years. b)determine the outstanding balance after the 24 th payment. How much of this payment was allocated to principal and how much to interest? c) Answer part b) for the 36 th and 60 th payments. d)what is the total of all the payments made over the 30 year period? How much was interest and how much was principal? 9 A car dealer presents two choices to a person seeking a particular car. Option A Purchase by making monthly payments of $ for 3 years, beginning a month from now. Option B Lease for 3 years by making monthly payments of $ (beginning a month from now) with an option to purchase at a price of $ at the end of the lease. Assuming that money yields 4% per annum compounded monthly after tax and inflation, calculate the cost of the car for each option. Which option is cheaper? Explain. Internet Exploration Check your answer to Exercise 6 using the Future Worth Calculator on This site offers a calculator to determine how much you must save to be a millionaire: Try a few calculations and check that it gives the correct answers. PART 2: ACTIVITY 3.5 EXERCISES 63
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