Chapter 5 Review Problems

Note: b. might explain how much an “average” smoker could save in 12 years (assuming a constant price for cigarettes and a fixed interest rate).

^[2] $500.00 is deposited at the end of every six months for nine years in an account paying 10.0% compounded semi-annually. Calculate the accumulated value of the deposits.

^[3] Calculate the amount of interest included in the accumulated value of $600.00 deposits made at the end of each month for 5 years. The interest rate is 13.5% compounded monthly.

^[4] Bill Holden is preparing retirement plans for his employees. He requires each employee to deposit $265.00 at the end of each month for 9 years. The interest rate is 8.75% compounded monthly.

1. How much money will be in each employee’s account at the end of 9 years?
2. How much will each employee have actually contributed?
3. How much of the amount will be interest?

^[5] Corinne Smith made $2,750 deposits every 6 months into a registered retirement savings plan paying 11.25% compounded semi-annually. Just after making the 16th deposit, the interest rate changed to 10.00% compounded quarterly. If neither deposits nor withdrawals were made during the next five years how much would Ms. Smith then have in her account?

^[6]  Calculate the present value of the ordinary annuity.

^[7] You wish to take two years off work to attend school and also wish to receive $950.00 at the end of every month for the 2  years. If you were able to deposit money into an account paying 10.00% compounded monthly:

1. How much should be deposited when you take the time off?
2. How much interest will you receive in the two years?

^[8] A laptop was bought by paying $750 down and an installment contract with payments of $85 at the end of each month for 2.5 years. If the interest was calculated at 16.9% compounded monthly:

1. What was the equivalent cash price of the laptop?
2. How much was the cost of the financing?

^[9] Peter Van Dusen opened a trust account to fund his son’s education. The account paid 10.25% compounded quarterly. His son is expected to require four years of quarterly payments of $2,000 with the first payment occurring 10 years 3 months from today . How much must Mr. Van Dusen deposit now so that his son will be able to receive the 4 years of payments? This is an example of a deferred annuity.

^[10] To purchase a new trawler-type yacht for chartering, Henry Skipper signed an agreement to borrow the entire amount and to make payments of $2,750 at the end of every month for seven years.

1. What was the purchase price of the yacht if money was worth 15.5% compounded monthly?
2. In his third year of operation, an economic downturn caused Mr. Skipper to miss payments 25 and 26. What payment was required at the time that payment 27 was due in order to bring the contract up to date?
3. Upon receipt of payment 27, the mortgage company wished to invoke a contractual clause and cancel the mortgage. How much (in addition to the payment calculated in part b above) would Mr. Skipper have to pay in order to fully pay out the mortgage?

^[11] Calculate the payment for each annuity:

^[12] A used Corvette can be bought for $15,000 cash or for equal payments at the end of each quarter for 5 years. Calculate the size of the quarterly payments at 10% compounded quarterly.

^[13] A gaming computer priced at $4,600 can be purchased for $1,900 down and the balance paid by 36 equal monthly payments at 13.8% compounded monthly. Calculate the size of the monthly payments.

^[14] Calculate the term of each annuity:

^[15] Charlie Horseshoe invested their $250,000 lottery winnings in a term deposit paying 8% compounded monthly for 10 years. For how long can $3,500 be withdrawn from the account at the end of each month starting at the end of the term deposit? Does your answer make sense? If not, why not?

^[16] $1,000 is deposited at the end of each month for 5 years. Find the nominal rate of interest compounded monthly at which the deposits will accumulate to $80,000. (Answer in % using 2 places of decimals.)

^[17] A camper van can be purchased for $27,500 plus 14% taxes and fees. The dealer will finance the balance owing after the payment of the sales tax and a down payment of 20% of the price (without the taxes and fees). Payments will be $330.30 at the end of each month for 7 years. What nominal rate of interest compounded monthly is being charged?

^[18] A computer valued at $2,800 can be bought for 25% down and monthly payments of $84.60 for 2.5 years. What effective rate of interest is being charged?

^[19] A loan of $30,000 is to be repaid with monthly payments over a period of 20 years. Calculate the total savings in interest if the loan is financed at 12.25% compounded monthly rather than 13% compounded monthly.

^[20] Judith Leisure-Lee is able to set aside $1,500 every 3 months from her income. She plans to buy a studio condominium at Hemlock Valley Ski Area when she has accumulated at least $50,000. How long will it take her if she can invest her savings at 9.5% converted quarterly? (State the answer in months.)

^[21] A firm believer in technological education wishes to provide an educational institution with $6,000 bursaries to be awarded at the end of each year for the next 10 years. If the institution can invest money at 8.75% effective, how much should the philanthropist donate, one year prior to the first award, to set up the fund for the 10 bursaries?

^[22] An agreement for sale contract carries payments of $4,500 at the end of every six months for 10 years. How much should you be willing to pay for the contract if you require a return of 12.25% compounded semi-annually on your money?

^[23] Samuel Hardy received $48,650 as a severance settlement when his position was terminated. Harvey had just celebrated his 41st birthday. He immediately and prudently invested the money in an account paying a guaranteed 9.6% compounded semi-annually until his 60th birthday last year. At that time, he converted the existing balance into an ordinary annuity paying $3,750 per month with interest at 10% compounded monthly. For how long will the annuity run until all the funds have been paid out?

^[24] Judith Leisure-Lee purchased a small studio condominium at Whistler for $115,000. She paid $40,000 down and agreed to make equal payments at the end of every month for 25 years. The interest rate was 13.25% compounded monthly.

1. What size payment is Judith making each month?
2. After 10 years of payments, how much will Judith still owe?
3.  How much will she have paid, in total, over the 25 years?
4. How much total interest will she have paid after 25 years of payments?

^[25] A well-used car, priced at $1,250 was sold on “easy terms” for a down payment of $250 and $50 per month for two years. What effective interest rate is being charged?

^[26] Dogwood Holdings financed a factory expansion by borrowing $325,000 at 10% compounded semi-annually for 10 years. Payments are to be made at the end of every six months.

1. Calculate the size of the payment.
2. How much of the 4th payment is interest?
3. Calculate the outstanding balance after the 4th payment.
4. Construct an amortization schedule for the first 4 payments.

^[27] Save-On-Auto Parts borrowed $120,000 to purchase a fleet of seven vans. They intend to repay by making monthly payments of $2,400. Interest is at 16% compounded monthly.

1. How many full payments will Save-On-Auto Parts have to make?
2. Calculate the size of the final payment, to be made one month after the last full $2,400 payment, which will fully amortize the debt.
3. How much interest is included in the 40th payment?
4. What percentage of the loan will have been repaid by the first 48 monthly payments?
5. How much total interest will be paid?

^[28] A New York lottery offers a choice to the winner of $1,000,000 cash or $8,500 per month for 15 years. Which alternative should the winner select if money is worth:

1. 6.0% compounded monthly?
2. 6.25% compounded monthly?

^[29] Find the present value and future value for an annuity whose periodic payments of $1,000.00 are made at the beginning of every quarter for 7 years. The rate of interest is 13% compounded quarterly.

^[30] Find the present value and future value for an annuity whose periodic payments of $3,000 are made at the beginning of each six months for 15 years. The rate of interest is 8.5% compounded semi-annually.

^[31] Amby Dextrous bought a grand piano, and agreed to a series of monthly payments of $225 for 4 years starting the date of sale. The rate of interest is 13% compounded monthly.

1. Calculate the cash price.
2. How much will be paid over the term of its financing?
3. How much of the total payments will be interest?

^[32] Find the cash value of a three year service contract for monthly payments of $1,600 paid at the beginning of each month. The value of money is 14% compounded monthly.

^[33] If an annuity with a present value of $50,000 has periodic payments at the beginning of each quarter for 10 years, find the size of the periodic payment. The rate of interest is 12% compounded quarterly.

^[34] An annuity whose present value is $50,000 is extinguished by payments of $650.00 made at the beginning of each month for 12 years.

1. Find the nominal rate of interest compounded monthly.
2. Calculate the effective rate.

^[35] Payments of $200 were made into a stock ownership plan at the beginning of each quarter for 15 years. They now have a net value of $50,000. What has been the nominal rate of return on the investment?

^[36] $100 is deposited into a retirement plan at the beginning of every month for 20 years. One month after the last deposit, money is withdrawn in equal monthly payments for 15 years. If interest is 8.5% compounded monthly, find the size of the monthly withdrawals.

^[37] If $75 is deposited into an education fund at the beginning of each month for 18 years and one month after the final deposit monthly withdrawals of $310.56 a month are made until the fund is exhausted, find the term of the annuity of withdrawals. Interest is 7.5% compounded monthly. (Try this problem both as an annuity due and an ordinary annuity.)

^[38] Find the present value of a deferred annuity whose periodic payment is $550 at the beginning of each year for 20 years, with the first payment following a two year period of deferment. The interest rate is 3.6% compounded annually.

^[39] Find the present value of a deferred annuity whose periodic payment of $360.00 is made at the end of every semi-annual period for 18 years after a deferment period of six months. The interest rate is 10% compounded semi-annually.

^[40] A deferred annuity has a present value of $15,000 and periodic payments are made at the beginning of each quarter for 10 years after a deferral period  of 8  years. The rate of interest is 6% compounded quarterly. Find the size of the periodic payment.

^[41] Tri-City Holdings borrows $500,000 to fund the expansion of the firm into its eastern market region. If the loan is to be repaid by making equal payments at the end of each quarter for 8 years beginning after a deferral period of 2 years, find the size of the periodic payments. Interest rate is 11% compounded quarterly.

^[42] Find the present value of a perpetuity whose periodic payments of $4,000 are made at the end of each quarter. Interest is at 6.8% compounded quarterly.

^[43] An institute lecturer position in Math of Finance is being funded by a perpetual fund . The fund earns interest at I0% compounded annually and is to pay $60,000 at the end of each year, with the first payment two years from the date of the fund being set up. Find the size of initial funding required.

^[44] You want to have accumulated $4,000 for your European trip four years from now. If interest is 6.4% compounded quarterly, find the size of quarterly deposits required for 4 years if deposits are made:

1. at the beginning of each quarter.
2. at the end of each quarter.

^[45] A three-year car lease has a present value of $16,600. If money is worth 13.5% compounded monthly, find the equivalent monthly lease payments, payable in advance for 3 years.

^[46] A Caribbean holiday tour package may be financed by making monthly payments of $300 at the beginning of each month for 2 years. Interest is 15% compounded monthly.

1. Find the purchase price now.
2. What will be the total paid in installments over the term?
3. How much interest will be paid over the term of the financing?

^[47] A building will produce net monthly incomes of $2,000 at the beginning of each month indefinitely. What is the maximum purchase price if one can get 14% compounded monthly on one’s money?

^[48] $400 is deposited into a retirement fund at the end of each quarter for 10 years and interest is paid at a rate of 9.6% compounded quarterly.

1. Find the accumulated balance at the end of the 10 years.
2. How much of that balance is interest?

Simple Ordinary Annuities, Annuities Due, General Annuities

^[52] Starting today, Mrs. Robinson will put $500 into her RRSP every month for 20 years. If her RRSP earns 6% compounded monthly, how much will she earn in interest over the 20 years?

^[53] Mrs. Watson wants to save $52,450 for a down payment on a house. She will save $2,000 per quarter, starting today.

1. If her invested funds earn 6% compounded quarterly, how long will it take her to reach her goal?
2. How much interest will she earn during that time?

^[54] You have just graduated from BCIT. You owe $15,238.98 in student loans. You will be charged 7% interest compounded monthly. You can afford to make monthly payments of $300 starting today.

1. How long (in years) will it take you to repay your student loans?
2. How much interest will you have paid?

^[55] How long will it take to save $100,000 if you start saving$1,597 every 3 months, starting today? Assume an interest rate of 6%, compounded quarterly.

^[56] You borrow $50,000 from the bank to consolidate your credit card debt and student loans. The bank charges you 12% interest, compounded monthly. The first payment is one month from now.

1. How long will it take to pay off the loan if you pay$504.25 per month?
2. What is the cost of financing?
3. How long will it take to pay off the loan if you pay only $500 per month?

^[57] Mr. Eskanderian contributes $1,000 into his RRSP at the end of every quarter for 10 years. If his RRSP earns 10% compounded quarterly, how much interest will he earn in the 10 years?

Deferred Annuities

^[58] Judith transfers $25,000 into an RRSP today. She plans to let the RRSP accumulate earnings at the rate of 8.75% compounded annually for exactly 10 years and then immediately purchase a 15-year annuity. The first withdrawal will start 3 months after she purchases the annuity. The annuity earns 9% compounded quarterly. What size of payment will she receive every 3 months?

^[59] Katherine has recently received an inheritance. She wants to set aside part of the inheritance to put it into an RRSP to save for her retirement. She anticipates that she will need to receive $1,200 per month for 15 years with the first withdrawal starting exactly 10 years from today. The invested funds will earn 7% compounded monthly for the entire 25 years. What amount must she contribute to her RRSP today?

^[60] Barry wants to set up an annuity that will pay him $3,000 per month for 20 years beginning when he turns 65 years of age. If his current age is 50 years and the invested funds will earn 6.5% compounded monthly, what amount must he invest today?

^[61] Samuel recently inherited money from his grandfather’s estate. He wants to purchase an annuity that will pay $5,000 every 3 months between age 60 (when he plans to retire) and age 65 (when his permanent pension will begin). The first withdrawal is to be 3 months after he reaches 60, and the last is to be on his 65th birthday. If Sam is currently 50.5 years old, and the invested funds will earn 6% compounded quarterly, what amount must he invest today?

^[62] It is time to start saving for your retirement. You are 40 years old and want to retire when you tum 60. You will deposit $1,000 into an RRSP at the beginning of every month for 20 years. You will then use the accumulated funds to purchase a 15-year annuity with the first withdrawal one month after your 60th birthday. Assume that the RRSP earns 7% compounded monthly, and the funds invested in the annuity earn 5% compounded monthly.

1. Find the size of the monthly withdrawals.
2. You decide that you will need at least $5,000 per month to live on when you retire at age 60. How much extra money must you contribute to your RRSP each month so you can withdraw $5,000 every month for 15 years?

^[63] Starting today, Giselle Lafleur will deposit $200 in her RRSP each month for 20 years. One  month  after  the  last  deposit, she will withdraw the money  in equal  monthly  withdrawals for 10 years.

1. Find the size of the monthly withdrawals if the invested funds earn j₁₂ = 9%.
2. How much interest will Miss Lafleur earn over the next 30 years?

^[64]  You have just celebrated your 20th birthday. You want to retire when you tum 65. Starting today , you are going to make monthly contributions to your RRSP, so that when you retire you can withdraw $2,250 per month for 15 years. The first withdrawal is made when you turn 65. You anticipate the invested funds will earn 7% compounded monthly.

1. How much money must you contribute each month to your RRSP to achieve your goal?
2. How much interest did you earn during the entire time?

^[65] Jean is thinking about retiring in five years. He would like to have $50,000 in an account when he retires. He decides to make monthly deposits (at the beginning of each month) in his local credit union where he can earn 7.0% compounded quarterly.

1. What would be the required monthly deposit to accumulate $50,000 in five years?
2. How much interest does John earn over the 5 years?

^[66] Six years from now, when you tum 55, you are planning to retire. You want to set aside some money today so you can receive $2,500 at the end of every quarter for 15 years with the first withdrawal 3 months after you tum 55. The invested funds earn 9% compounded semi-annually.

1. What amount must you invest today?
2. How much interest will you earn during the 21 years?

Perpetuities and General Annuities

^[67] You have become wealthy beyond your wildest dreams and would like to create a scholarship at BCIT. You would like to give $2,000 per year to a student studying business math. You would like your scholarship to continue in perpetuity.

1. How much money should you set aside today if the first payment is in one year and the interest rate is 8% compounded annually?
2. How much should you set aside if the first scholarship is today?

^[68] You take out a loan to buy a black Honda S2000 convertible sports car with leather interior . The bank requires that you put $9,000 down followed by payments of $925 at the end of every month for 5 years.

1. If the bank charges you 12% interest compounded monthly, what is the selling price of your car?
2. If the bank charges you 12% interest compounded semi­ annually, how much would your monthly payments be? Note: You borrow the same amount as found in part (a).

^[69] Mr. Bean borrows $7,500 today. He will repay the loan with quarterly payments at the end of every three months for three years. The interest rate charged is 9% compounded monthly. Find the size of the payment.

^[70] A corporation donates $11,000 to Langara. The funds are invested at 10% compounded annually per year. The interest is paid out each year as a scholarship. How much will be paid out each year if the first scholarship is paid immediately after the donation is received? Note: It’s easier to use the calculator.

^[71] You have just won the Set for Eternity Lottery; the lottery will pay you and your descendants $5,000 per month forever with the first payment in one month . Instead of receiving $5,000 per month forever you would like to receive the cash now. What is the cash value of the prize if the interest rate is j₁₂  = 6%?

^[72] The TRIUMF research lab at UBC recently received a donation from a private individual. The funds were matched two-to-one by the government. Each year they plan to pay out a scholarship of $8,596.59 as they anticipate earning 11% compounded quarterly on the invested funds. What amount did the private individual donate? (The first scholarship will be one year later). Round answer to the nearest dollar.

^[73] You have decided to purchase preferred shares of Plutonium Fuel Cells Inc. that pays a semi-annual dividend of $1.25 per share.

1. What would you be willing to pay per share if you want to earn at least 5% compounded semi-annually on your investment and the next dividend is to be paid in 6 months?
2. You are short of cash and need to sell your shares of Plutonium Fuel Cells Inc. Unfortunately, interest rates have risen to 8% compounded semi-annually. Calculate your gain or loss per share if the next dividend is due in six months – and the dividend per share remains the same.

^[74] You purchase 500 preferred shares of Yardmucks Coffee that pays a quarterly dividend of $0.26 per share. The next dividend is due in 3 months. Current interest rates are 8% compounded quarterly.

1. What would you be willing to pay for the 500 shares?
2. If interest rates drop to 6.5% compounded quarterly, how much would you expect to gain or lose if you sell all of your shares? Note: the next dividend is due in 3 months and the dividend per share is still $0.26.

^[75] The Winfall lottery offers you two choices for its grand prize. Either a cash prize of $1,500,000 today or $7,000 per month forever (with the first installment one month from now).

1. Which choice should you select if interest is 6% compounded monthly?
2. You choose the $1,500,000 today and deposit it into an account earning 6% compounded monthly. How much could you withdraw each month forever? Assume the first withdrawal is one month from now.

^[76] You borrow $50,000 and agree to make monthly payments for 15 years starting one month later. Calculate the size of your payments if the interest rate is 9% effective.

^[77] You borrow $20,000 today from a moneylender called, The Money Branch. As a result of your bad credit The Money Branch charges you an interest rate of 28.8%. You will repay the loan with equal monthly payments over four years with the first payment one month from now.

1. Find the size of the monthly payment if the interest is: (i)28.8% compounded monthly or (ii) 28.8% compounded semi-annually.
2. How much less interest would you pay if they compound the interest semi-annually, instead of monthly?

^[78] The University of Edmonton received a donation from a wealthy individual. Some of the donated money will be set aside to create a scholarship fund that will pay out $10,000 at the end of every 6 months, in perpetuity. If the invested funds can earn 8% compounded semi-annually, instead of 5% compounded semi-annually, how much less money must they set aside today to pay out a $10,000 scholarship?

^[79] You purchase 200 preferred shares of Black Bear Brewing that pays a dividend of $1 per share every 3 months. Current interest rates are 12% compounded monthly. The next dividend is due in 3 months.

1. How much would you be willing to pay for the 200 shares?
2. If interest rates fall to 8% compounded quarterly, how much would you expect to gain or lose if you sell all of your shares? Note: the next dividend is due in 3 months and the dividend per share is unchanged.

Mortgages

^[80] You take out a $200,000 mortgage:

1. What is the periodic interest rate per month if the rate is
   1. 12% compounded monthly?
   2. 12% compounded semi-annually?
2. If you borrow $200,000, how much interest is paid in the first month? Use both rates and compare your answers.

^[81] A debt of $1,200 is repaid by monthly payments of $350 at the end of each month. The interest rate is 12% compounded monthly. Construct the complete amortization schedule without using the AMRT keys. Then go back and verify all the answers using the AMRT and  Pl/P2 keys .

^[82] You purchase a house in Surrey for $220,000 and put 25% down so you could receive a conventional mortgage. You decide to get your mortgage at Superstore since they give you free groceries each year as an incentive. A 5-year term mortgage (i.e., the interest rate is fixed for 5 years) is negotiated with Simplii Financial in which the balance is amortized over 20 years (repaid with equal payments over 20 years) at 6.70% interest compounded semi-annually.

1. Calculate your monthly payment. The bank rounds up the payment to the next dollar.
2. COMP n to verify that n is slightly smaller than 240. If the value of n = -100 then you forgot to make the payment negative. If the value of n is exactly 240 then you forgot to re-enter the payment.
3. Find out how the first payment is distributed between interest and principal. Compare this to the results for the 60th payment.
4. How much interest did you pay in the first year? By how much was the balance outstanding reduced in the first year?
5. What percent of the original mortgage was paid off in the first two years?
6. How much principal was paid off in the first five years? How much interest did you pay in the first five years?
7. How much interest did you pay in the fifth year of the mortgage? How much principal was repaid in the fifth year?
8. How much will you still owe after you have made five years of payments?
9. Calculate the value of the final payment assuming that the interest rate never changes during the 20 years.

^[83] The Archibald’s are eligible for a Canada Mortgage and Housing Corp. insured mortgage allowing them to qualify for a mortgage of up to 95% of the selling price of the house. They are also subject to the 30% rule: no more than 30% of their gross income can go towards paying the mortgage and property taxes.

1. What is the maximum mortgage they qualify for if their gross monthly income is $5,000 and they want to amortize the mortgage over 25 years? Assume that the property taxes on the house are $1,800 per year (after the home owner’s grant). Current mortgage rates are 6.80% compounded semi-annually. Round answer to the nearest $100.
2. The Archibald’s take out a mortgage for $195,000 with Citizen’s Bank amortized over 25 years at 6.8% interest compounded semi-annually for a 5-year term. What is the size of the monthly mortgage payment(round up to the next dollar)?
3. How much interest did they pay in the first five years of the mortgage?
4. How much money would they still owe on this mortgage after five years of payments?
5. When they renew their mortgage in five years time, mortgage rates have fallen to 6.0 % compounded semi­ annually for a five-year term. They have saved $15,000 and will use it to reduce the size of the mortgage. Find the size of their new monthly payments assuming the balance outstanding is amortized over the remaining time. The bank rounds up the payment to the next dollar.
6. What is the size of the final payment of the renewed mortgage assuming that the interest rate does not change during the remaining 20 years?

^[84] Banks normally use the 30% rule: no more than 30% of your gross income can go towards paying your mortgage and property taxes.

1. If your gross monthly income is $4,000 per month and your property taxes are $2,400 per year (after the $470 home owners grant), what is the largest mortgage a bank would authorize if the mortgage is amortized over 25 years and the rate of interest is 7.45% compounded semi-annually. Round answer to the nearest $100. (Assume monthly payments).
2. You purchase a fixer-upper house in downtown Mission for $164,000. Your down payment is 25% and you negotiate a first mortgage for the balance. The mortgage is at 7.45%, compounded semi-annually, amortized over 25 years for a 3-year term. How large a monthly payment is required? The bank rounds payment up to the next dollar.
3. How much interest did you pay in the first three years of the mortgage?
4. What percent of the original mortgage have you paid off in the first three years?
5. How much interest did you pay in the third year only?
6. In three years the term of your mortgage is up and you wish to renew. Interest rates have increased to 9.25% compounded semi-annually for a three-year term mortgage. At this time you make a lump-sum payment of $10,000 to reduce the size of your mortgage. Calculate the size of the new monthly payments assuming the balance outstanding is amortized over the remaining time. Round up to the next dollar.
7. What is the size of the final payment of the renewed mortgage assuming that the interest rate does not change during the remaining 22 years?

^[85]  Suppose you take out a mortgage amortized over 25 years for $179,940 at j2 = 12.5%.

1. Find the size of the monthly payment. Round payment to nearest penny.
2. How much time and money would you save if you make 26 bi-weekly payments (equal to half of the monthly payment) instead of twelve monthly payments each year? Assume that the interest rate never changes during the 25 years.

^[86]       In March 2012 the Beckers purchased a house in Delta for $512,500. They made a down payment of exactly 20% and took out a mortgage with TD Canada Trust for the balance at an interest rate of 7.5% compounded semi-annually, for a 5-year term, amortized over 25 years.

1. What is the size of the monthly payment? The bank rounds the payment up to the next dollar.
2. How much of the 30th payment was interest?
3. How much interest did the Beckers pay in the 3rd year of the mortgage?
4. Today, (March 2017), they have decided to increase the size of their mortgage and use the money for house renovations. How much extra money can they borrow if they want to keep the same monthly payment as before but still pay off the mortgage by March 2037 (twenty years from now)? The interest rate has fallen to 5.1%, compounded semi-annually for a five-year term.
5. The Beckers increase their mortgage to $450,000 and amortize it over 20 years at 5.1%, compounded semi­ annually. What is the size of their monthly payment? (The bank rounds up the payment to the next dollar.)
6. What is the size of the final payment assuming that the interest rate stays the same over the remaining twenty years?

^[87] The Blacks are considering purchasing a three bedroom townhouse in the Killarney area of Vancouver. Their gross monthly income is $12,000 per month. The property taxes on the townhouse are $3,000 per year, and they also have to pay a monthly maintenance fee of $150 per month for the upkeep on the townhouse complex. Banks normally use the 30% rule: no more than 30% of your gross income can go towards paying your mortgage, property taxes, and monthly maintenance fees. What is the largest mortgage a bank would authorize if the mortgage is amortized over 25 years and the rate of interest is 5.6%, compounded semi-annually? (Assume monthly payments.)

^[88] Five years ago the Smiths purchased a home in North Vancouver for $650,000. They made a down payment of exactly 20% and mortgaged the balance with Westminster Savings Credit Union. The interest rate was 5.6%, compounded semi-annually, for a 5-year term, amortized over 25 years.

1. Calculate the size of the monthly payment required. The credit union rounds the payment up to the next dollar.
2. What percentage of the original mortgage was paid off in the first 3 years?
3. How much interest did the Smiths pay in the fourth year of the mortgage?
4. The Smiths, after having made 5 years of payments, made a lump-sum payment to reduce the outstanding balance to $450,000. What was the amount of the lump-sum payment?
5. After making the lump sum payment, the Smiths renew their mortgage for another 5-year term, amortized over the remaining time, at 6.8%, compounded semi-annually. Calculate the new monthly payment. Round the payment up to the next dollar.
6. Assuming the interest rate remains the same over the remaining time, what is the size of the final payment?

^[89] In October 2012 the Reids obtained a mortgage for $380,000 at 6.8% interest compounded semi-annually, for a 5-year term, amortized over 25 years. (The bank rounds the payment up to the next dollar.) Today, (October 2017), they have decided to increase the size of their mortgage and use the money for house renovations. How much more money can they borrow if they want to keep the same monthly payment as before but still pay off the mortgage by October 2037 (twenty years from now)? The interest rate has fallen to 4.3%, compounded semi-annually for a five-year term.

^[90] You purchase a new car. The dealer offers you terms of 20% down and the remainder financed over five years at an interest rate of 8% compounded monthly.

1. 1. Find the size of your monthly payment if your first payment is due at the end of the month and the price of the car was $33,906.25 including GST, PST, documentation fee, and government environmental levies.
   2. What is the cost of financing (how much interest will you pay over the life of the loan)?

^[91] You would like to save for your retirement by making monthly deposits of $200 into an account. What nominal interest rate of interest, compounded monthly, must you earn to accumulate $1,000,000 in thirty years? Assume that the first payment will be at the end of the month.

^[92] You are considering quitting smoking due to the high cost of a pack of cigarettes. You smoke 1 pack a day at a cost of $7.50. If you put the $7.50 you would have spent on cigarettes, into a savings account earning 5.75% interest compounded daily, how much would you have in the bank at the end of ten years?

^[93]  Upon graduation, you have a student loan of $15,000. The most you can afford to pay is $550 per month. How long will it take you to repay the loan with payments of $550 per month starting in one month if the interest rate is 6% compounded monthly?

^[94] Kent sold his car to Carolyn for $1,000 down and monthly payments of $120.03 at the end of every month for 3.5 years. The interest rate charged is 12%, compounded monthly. What was the selling price of the car?

^[95] Rajinder bought a car with $5,000 down followed by equal monthly payments of $783.41 at the end of every month for 2 years at 16% compounded monthly.

1. What is the selling price of the car?
2. What is the cost of financing?

^[96] Manpreet paid $14,000 to buy a used car. He made a down payment followed by equal monthly payments of $249.50 at the end of every month for 4 years at 8% compounded monthly.

1. What is the size of the down payment?
2. What is the cost of financing?

^[97] For $42,000 an individual can purchase a 5-year annuity from Continental Life and receive monthly payments of $871.85 for 5 years with the first payment one month from now. What effective rate of interest does this investment earn?

Deferred Perpetuities

^[98] A scholarship fund is to be set up. The fund will pay out a scholarship of $20,000 every year with first scholarship paid out two years after the fund is set up. Find the size of the donation needed. Assume the fund will earn 10% effective.

^[99] A bursary fund for BCIT honor students is to be funded by a perpetual fund. The fund earns interest at 10% compounded annually and is to pay $30,000 each year, with the first payment four years after the fund is set up. Find the size of the initial funding that is required.

^[100] An alumnus wants to donate a sum of money to his Alma Mater that will provide a scholarship of $750.00 every six (6) months in perpetuity. If money can be invested at 6% compounded semi-annually and the first $750.00 is to be awarded at the end of one year, how much must he donate to the school today?

^[101] You are considering purchasing shares of New Wave Technology Corp. The company has stated that they will pay dividends of $0.72 per share every three (3) months with the first dividend paid exactly four (4) years from today. If the current interest rates are 8% compounded quarterly, what would you be willing to pay for one share today?

Using All 5 Keys

^[102] You just celebrated your 40th birthday. You dream about retiring when you tum 55. You currently have $80,000 accumulated in your retirement plan. You decide to make deposits each month into a retirement plan for exactly 15 years, starting today. You want to purchase an annuity which will pay you $6,000 per month for l 0 year, with the first withdrawal starting one month after your 55th birthday. The retirement plan and the annuity earn 6% compounded monthly.

1. How much must you deposit each month into your retirement plan?
2. How much interest will you earn over the ENTIRE 24 years?

^[103] Barney just celebrated his 40th birthday . He currently has $52,034 accumulated in his retirement plan and he plans to continue making equal monthly deposits into a savings account for 15 years, starting today. Two months after his last deposit, he intends to withdraw $4,000 per month for his living expenses for a period of 5 years. The invested funds earn 6%, compounded monthly, for the entire 20 years.

1. Find the size of the monthly deposits.
2. How much interest will he earn over the entire 20 years?

^[104] You have $50,000 in your RRSP today. For the next 12.5 years you will contribute $500 per month into your RRSP with the first deposit made one month from now. How much will you have in your RRSP at the end of 12.5 years if you earn 6.9598% compounded monthly on your RRSP?

^[106] Starting today, you will contribute $695.09 per month into an RRSP for a period for 5 years. You will then use the accumulated funds to purchase an annuity with monthly payments paid out over 12.5 years. What will be the size of the monthly withdrawals if the first withdrawal is made 2 months after the last deposit? Assume you earn 6.9598% compounded monthly the entire time.

^[107] What amount will be in an RRSP after 20 years if monthly contributions of $300 are made for the first 15 years and then contributions of $600 per month are made for the subsequent 5 years? The first deposit is made one month from now and the funds invested in the RRSP earn 7% compounded monthly.

^[108] Starting today, you contribute $1,200 every 3 months into your RRSP for five years. The interest rate was 10% compounded quarterly for the first 2 years and 9% compounded quarterly for the last 3 years. How much will you have in your RRSP at the end of 5 years?

^[109] Herb has made contributions of $2,000 to his RRSP at the end of every 6 months for the past 8 years. The RRSP has earned 9.5% compounded semi-annually. Today, he moved the funds to a different play, paying 8% compounded quarterly. He will now contribute $1,500 at the end of every 3 months. How much will he have in his RRSP 7 years from now?

^[110] The Alumni Association of BCIT would like to set up a scholarship that will pay out $2,000 every 6 months forever. The funds will be deposited in an account which earns 8.16% effective. The first scholarship will be awarded 2 years after the funds are deposited. How much money do they need to set aside today?

^[111] You plan to contribute $900 every 6 months into your retirement plan for a period of 15 years. How much INTEREST would you earn over the 15 years if your RRSP earns 7% compounded monthly? The first contribution is made 6 months from now.

^[112] Barney just celebrated his 25th birthday. Starting today he will make contributions every month into his retirement plan for a period of 30 years. How much must Barney contribute every month to his retirement plan so he can withdraw $5,000 per month for a period of 20 years with the first withdrawal starting two months after his last deposit? Assume Barney earns 8.5%, compounded quarterly, the entire time.

^[113] Marika has already accumulated $18,000 in her RRSP . If she contributes $2,000 at the end of every 6 months for the next 10 years, and $300 per month for the subsequent 5 years, what amount will she have in her plan at the end of the 15 years? Assume that her plan earns 9%, compounded semi­ annually, for the first 10 years and 9% compounded monthly for the next 5 years.

^[114] How much larger will the value of an RRSP be at the end of 25 years if the contributor makes month-end contributions of $300 instead of year-end contributions of $3,600? In both cases the RRSP earns 8.5%, compounded semi-annually.

^[115] What will be the amount in an RRSP after 25 years if contributions of $2,000 are made at the beginning of each year for the first 10 years and contributions of $4,000 are made at the beginning of each year for the subsequent 15 years? Assume that the RRSP earns 8%, compounded quarterly.