Deck 11: Annuities: Periodic Payment, Number of Payments, and Interest Rate

Question 1

Four years from now, Tim and Justine plan to take a year's leave of absence from their jobs and travel through Asia, Europe, and Africa. They want to accumulate enough savings during the next four years so they can withdraw $3,000 at each month-end for the entire year of leave. What amount must they pay into the fund at the end of every calendar quarter for the next four years to reach their goal? The planning assumptions are that their savings will earn 6% compounded quarterly for the next four years and 4.2% compounded monthly during the fifth year.

Answer 1

$1,962.61

Question 2

Howardson Electric obtained a $90,000 loan at 9.75% compounded monthly. What size of semi-annual payments will repay the loan in 10 years?

Answer 2

$7,206.60

Question 3

What payments must be made at the end of each quarter to an RRSP earning 4.2% compounded annually so that its value 8½ years from now will be $15,000?

Answer 3

$370.42

Question 4

Aden is scheduled to make a lump payment of $25,000, 11 months from now, to complete a real estate transaction. What end-of-month payments for the next 11 months should the vendor be willing to accept instead of the lump payment if he can invest the funds at 5.4% compounded monthly?

Question 5

Dr. Collins wants the value of her RRSP 30 years from now to have the purchasing power of $500,000 in current dollars.
a) Assuming an inflation rate of 2% per year, what nominal dollar amount should Dr. Collins have in her RRSP after 30 years?
b) Assuming her RRSP will earn 8.5% compounded semi-annually, what contributions should she make at the end of every three months to achieve the goal?

Question 6

The interest rate on a $200,000 loan is 8% compounded quarterly.
a) What payments at the end of every quarter will reduce the balance to $150,000 after 3½ years?
b) If the same payments continue, what will be the balance seven years after the date that the loan was received?
c) How much interest will be paid during the first seven years?

Question 7

The interest rate on a $30,000 loan is 4.5% compounded monthly. What end-of-month payments are required to pay off the loan in eight years?

Question 8

Suppose you choose to pay off a loan over 10 years instead of 5 years. The principal and interest rate are the same in both cases. Will the payment for the 10-year term be: (i) Half the payment for the 5-year term? (ii) More than half the payment? or (iii) Less than half the payment? Give the reasoning for your choice.

Question 9

Calculate the amounts that must be invested at the end of every six months at 4.75% compounded semi-annually in order to accumulate $500,000 after 20 years.

Question 10

Calculate the amount that must be invested at the end of each year at 5% compounded annually in order to accumulate $500,000 after
a) 25 years. b) 30 years.

Question 11

In order to accumulate $500,000 after 25 years, calculate the amounts that must be invested at the end of each year at
a) 6% compounded annually. b) 8% compounded annually.

Question 12

The interest rate on a $100,000 loan is 7.5% compounded quarterly.
a) What quarterly payments will reduce the balance to $75,000 after five years?
b) If the same payments continue, what will be the balance 10 years after the date that the loan was received?

Question 13

What monthly payments for 15 years will pay off a $50,000 loan at 8.25% compounded monthly?

Question 14

Brenda and Tom want to save $30,000 over the next four years for a down payment on a house. What amount must they regularly save from their month-end pay if their savings can earn 5.5% compounded semi-annually?

Question 15

Mr. Braun wants the value of his RRSP 25 years from now to have the purchasing power of $400,000 in current dollars.
a) Assuming an inflation rate of 4% per year, what nominal dollar amount should Mr. Braun have in his RRSP after 25 years?
b) What contributions should he make at the end of every three months to achieve the goal if his RRSP earns 5.5% compounded semi-annually?

Question 16

In order to purchase another truck, Beatty Transport recently obtained a $50,000 loan for 5 years at 5.8% compounded semi-annually. The loan is structured to reduce the balance owing to $10,000 at the end of the five-year period. How much are Beatty's end-of-month loan payments?

Question 17

Henrick can buy a rural property from a family member for $700,000 with terms of $100,000 down and the balance payable over 20 years by quarterly payments including interest at 8% compounded annually. What will be the size of the payments? How much interest will Henrick pay over the life of the loan?

Question 18

Ken and Belinda have two children, aged 3 and 6. At the end of every six months for the next 12½ years, they wish to contribute equal amounts to a Registered Education Savings Plan (RESP). Six months after the last RESP contribution, the first of 12 semi-annual withdrawals of $5,000 will be made. If the RESP earns 8.5% compounded semi-annually, what must be the size of their regular RESP contributions?

Question 19

Beth and Nelson want to accumulate a combined total of $600,000 in their RRSPs by the time Beth reaches age 60, which will be 30 years from now. They plan to make equal contributions at the end of every six months for the next 25 years, and then to make no further contributions for the subsequent five years of semi-retirement. For planning purposes, assume that their RRSPs will earn 7% compounded semi-annually for the next 30 years.
a) What should be their combined semi-annual RRSP contributions?
b) What combined monthly amount can they expect if they use the $600,000 in their RRSPs 30 years from now to purchase 25-year ordinary annuities? Assume that the funds used to purchase the annuities will earn 7.2% compounded monthly.

Question 20

John has $100,000 available to purchase an annuity. What end-of-month payments can he expect if the funds earn 5.4% compounded monthly and the payments run for
a) 10 years? b) 20 years?

Question 21

A conditional sale contract for a $1,450 transaction required a 10% down payment with the balance to be paid by 12 equal monthly payments. The first payment is due six months after the date of the purchase. The retailer charges an interest rate of 13% compounded semi-annually on the unpaid balance. What is the monthly payment?

Question 22

Calculate the term, expressed in years and months, of the following ordinary annuity.

Question 23

Suppose that you contribute $425 per month to your RRSP. Rounding up to the nearest month, how much longer will it take for the RRSP's value to reach $500,000 if it earns 4.2% compounded annually than if it earns 4.2% compounded monthly?

Question 24

How long will $500,000, in an investment account that earns 5.25% compounded monthly, sustain month-end withdrawals of $3,000?

Question 25

Rounded to the next higher month, how long will it take end-of-month deposits of $500 to accumulate $100,000 in a savings account that pays interest of 5.25% compounded monthly?

Question 26

How much longer will it take month-end RRSP contributions of $500 to accumulate $500,000 than month-end contributions of $550? Assume that the RRSP earns 3.75% compounded monthly. Round the time required in each case to the next higher month.

Question 27

How much longer will it take monthly payments of $1,000 to pay off a $100,000 loan if the monthly compounded rate of interest on the loan is 5.25% instead of 4.875%?

Question 28

Farah has $600,000 in her RRSP and wishes to retire. She is considering using the funds to purchase an annuity that earns 5% compounded annually and pays her $3500 at the end of each month. If she buys the annuity, for how long will she receive payments?

Question 29

Calculate the term, expressed in years and months, of the following ordinary annuity.

Question 30

Calculate the term, expressed in years and months, of the following ordinary annuity.

Question 31

If $300,000 is used to purchase an annuity earning 4.5% compounded monthly and paying $2,500 at the end of each month, what will be the term of the annuity? Include the final, smaller annuity payment in the total.

Question 32

Noreen's RRSP is currently worth $125,000. For the next 10 years, she will make contributions at the end of every six months. How much does Noreen need to contribute at the end of every six months for the next 10 years to reach her goal of having a total of $500,000 when she retires? Her RRSP earns 8% compounded annually.

Question 33

Using the "Cool Million" Chart An interactive "Cool Million" chart in the textbook's OLC enables you to visualize the growth of your retirement savings over several years. Another feature allows you to determine what changes you would need to make to your investment plan for you to attain nominal "millionaire" status on your 65^th birthday. Go to the Student Edition on this textbook's OLC. In the navigation bar, select "Chapter 11" in the drop-down box. In the list of resources for Chapter 11, select "Links in Textbook" and then click on the link named "Cool Million." Enter data for an investment plan that will be reasonable for you after you gain fulltime employment. Note that the "Expected rate of return" and the "Expected inflation rate" are both annually compounded rates. Click on the "Calculate" button to generate a new chart. The series of blue bars show the growth of your investments. If you move your cursor over any bar, the numerical value (in $000s) will be displayed. The series of purple bars represent the inflation-adjusted value, or purchasing power, of the investments in terms of dollars at the very beginning-the "Your age" date. If you click on the "View Report" button, a window containing a bulleted list of three suggested changes will appear. Any one of these changes to your savings plan will enable you to accumulate (a nominal) $1,000,000 at your target retirement age. Adjust one or more items in your input data to arrive at a plan that represents your best chance of entering retirement as a millionaire. What will be the purchasing power of your nominal $1,000,000 in beginning dollars?

Question 34

What duration of annuity paying $5,000 at the end of every quarter can be purchased with $200,000 if the invested funds earn 5.5% compounded semi-annually?

Question 35

Calculate the term, expressed in years and months, of the following ordinary annuity.

Question 36

You intend to accumulate $100,000 in 10 years instead of 20 years by making equal monthly investment contributions. Will the monthly contribution for a 10-year plan be: (i) Twice the monthly contribution for a 20-year plan? (ii) Less than twice the monthly contributions? or (iii) More than twice the monthly contributions? Assume the same rate of return in both cases. Give the reasoning for your choice.

Question 37

Sofia has been approved an RBC Royal Bank four-year $25,000 car loan at 7.5% compounded monthly. What end-of-month payments will reduce the balance on the loan after four years to the expected trade-in value of $4000,

Question 38

By the time he turns 60, Justin (just turned age 31) wants the amount in his RRSP to have the purchasing power of $250,000 in current dollars. What annual contributions on his 32^nd through 60^th birthdays inclusive are required to meet this goal if the RRSP earns 4% compounded annually?

Question 39

Calculate the term, expressed in years and months, of the following ordinary annuity.

Question 40

Rashid wants to use $500,000 from his RRSP to purchase an annuity that pays him $2,000 at the end of each month for the first 10 years and $3,000 per month thereafter. Global Insurance Co. will sell Rashid an annuity of this sort with a rate of return of 4.8% compounded monthly. For how long will the annuity run?

Question 41

Harold's RRSP is already worth $56,000. Rounded to the next highest month, how long will it take the RRSP to reach $250,000 if additional contributions of $2,000 are made at the end of every six months? Assume the RRSP earns 3.9% compounded monthly.

Question 42

Georgina is about to retire with $188,000 in her RRSP. She will use the funds to purchase an annuity providing payments of $6000 at the end of each quarter. What will be the annuity's term if the funds invested in the annuity earn 4.8% compounded monthly?

Question 43

Calculate the nominal and effective rate of interest for the following ordinary annuity. Determine the nominal interest rate whose compounding interval equals the payment interval.

Question 44

If you contribute $250 per month to an RRSP instead of $500 per month, will the time required to reach a particular savings target be (pick one): (i) Twice as long? (ii) Less than twice as long? (iii) More than twice as long? Give the reason for your choice.

Question 45

An annuity purchased for $175,000 pays $4,000 at the end of every quarter. How long will the payments continue if the funds earn 7% compounded semi-annually?

Question 46

The interest rate on a $100,000 loan is 9% compounded monthly. How much longer will it take to pay off the loan with monthly payments of $1,000 than with monthly payments of $1,050?

Question 47

Calculate the nominal and effective rate of interest for the following ordinary annuity. Determine the nominal interest rate whose compounding interval equals the payment interval.

Question 48

Calculate the nominal and effective rate of interest for the following ordinary annuity. Determine the nominal interest rate whose compounding interval equals the payment interval.

Question 49

Bonnie and Clyde want to take a six-month leave of absence from their jobs to travel extensively in South America. Rounded to the next higher month, how long will it take them to save $40,000 for the leave if they make month-end contributions of $700 to their employer's salary deferral plan? The salary deferral plan earns 5.5% compounded semi-annually.

Question 50

How long will it take monthly payments of $400 to repay a $50,000 loan if the interest rate on the loan is 8% compounded semi-annually?
b) How much will the time to repay the loan be reduced if the payments are $40 per month larger?

Question 51

Ernie's Electronics sells an LED HD television priced at $2395 for $100 down and payments of $100 per month, including interest at 9.6% compounded monthly. How long after the date of purchase will the final payment be made?

Question 52

If you double the size of the monthly payment you make on a loan, will you pay it off in (pick one): (i) Half the time? (ii) Less than half the time? (iii) More than half the time? Give the reason for your choice.

Question 53

Calculate the nominal and effective rate of interest for the following ordinary annuity. Determine the nominal interest rate whose compounding interval equals the payment interval.

Question 54

The McGowans are arranging a $220,000 mortgage loan from their bank. The interest rate on the loan will be 4.9% compounded semi-annually.
a) What will the monthly payments be if the loan has a 20-year term?
b) If the McGowans choose to pay $1500 per month, how long will it take to pay off the loan?

Question 55

A 70-year-old male can purchase either of the following annuities for the same price from a life insurance company. A 20-year-term annuity will pay $394 at each month-end. A life annuity will pay $440 at the end of each month until the death of the annuitant. Beyond what age must the man survive for the life annuity to have the greater economic value? Assume that money can earn 3.3% compounded monthly.

Question 56

How much sooner will a $65,000 loan at 7.2% compounded monthly be paid off if the monthly payments are $625 instead of $600? What will be the approximate saving in total interest costs over the life of the loan?

Question 57

A 65-year-old male can purchase either of the following annuities from a life insurance company for $50,000. A 25-year term annuity will pay $307 at the end of each month. A life annuity will pay $408 at the end of every month until the death of the annuitant. To what age must the man survive for the life annuity to have the greater economic value? Assume that money can earn 6% compounded monthly.

Question 58

a) How long will it take monthly payments of $600 to repay a $65,000 loan if the interest rate on the loan is 9.5% compounded semi-annually?
b) How much will the time to repay the loan be reduced if the payments are $50 per month larger?

Question 59

If $400,000 accumulated in an RRSP is used to purchase an annuity earning 3.9% compounded monthly and paying $3,500 at the end of each month, what will be the term of the annuity?

Question 60

Twelve years ago, Mr. Lawton rolled a $17,000 retiring allowance into an RRSP that subsequently earned 7% compounded semi-annually. Three years ago he transferred the funds to an RRIF. Since then, he has been withdrawing $1,000 at the end of each quarter. If the RRIF earns 4.2% compounded quarterly, how much longer can the withdrawals continue?

Question 61

Rounding up to the nearest month, for how long will a $100,000 fund sustain beginning-of-the-month withdrawals of $700 if the fund earns 6% compounded monthly?

Question 62

Another type of sales promotion for vehicles is to advertise the choice between a "Cash Purchase Price" or "0% Purchase Financing." The tiny print at the bottom of a GM Canada full-page advertisement included the statement: "The GMAC purchase finance rates are not available with and are not calculated on the 'Cash Purchase Price' shown. The difference between the price for the GMAC purchase finance offer and the 'Cash Purchase Price' offer is deemed under provincial laws to be a cost of financing." In other words, there are two prices for a vehicle-a lower price if you pay cash and a higher price if you want to take advantage of the "0% financing." An additional disconcerting aspect of this type of promotion is that the higher price for the 0% financing is usually not quoted in the advertisement. Rather, it must be negotiated with the dealer.
Suppose the Cash Purchase Price of a car is $23,498, and the price that qualifies for full 0% financing (with 48 monthly payments) turns out to be $26,198. What effective interest rate will you be paying for the "0% financing?"

Question 63

An advertisement for new vehicles offered "1.9% 12-month financing or $2,000 cash back." A truck buyer financed $17,000 at the low interest rate instead of paying $15,000 cash (after the $2,000 rebate). What was the effective rate of interest on the loan if the foregone cash rebate was treated as part of the cost of financing? (The 1.9% interest rate was a monthly compounded nominal annual rate.)

Question 64

How much must be invested at the beginning of each year in order to accumulate $750,000 after 25 years? The invested funds earn 6% compounded annually.

Question 65

An advertisement for Ford trucks offered "2.9% financing (for 48 months) or $2,000 cash back." A truck buyer financed $20,000 at the low interest rate instead of paying $18,000 cash (after the $2,000 rebate). What was the effective rate of interest on the loan if the foregone cash rebate is treated as part of the cost of financing? (The 2.9% interest rate is a monthly compounded nominal annual rate.)

Question 66

If $100,000 will purchase a 20-year annuity paying $830 at the end of each month, what monthly compounded nominal rate and effective rate of interest will the invested funds in the annuity earn?

Question 67

If $100,000 will purchase a 20-year annuity paying $739 at each month's end, what monthly compounded nominal rate and effective rate of interest are earned by the funds?

Question 68

After 10½ years of contributions of $2,000 at the end of every six months to an RRSP, the accumulated amount stood at $65,727.82. What semi-annually compounded nominal rate and effective annual rate of return were earned by the funds in the RRSP?

Question 69

A major daily newspaper charges $260 (paid in advance) for an annual subscription, or $26 per month payable at the end of each month to the carrier. What is the effective interest rate being charged to the monthly payment subscribers?

Question 70

$2,000 will be contributed to an RRSP at the end of every six months for 20 years. What effective rate of return must the funds in the plan earn if it is to be worth $250,000 at the end of the 20 years?

Question 71

What beginning-of-month withdrawals can a $400,000 RRIF (Registered Retirement Income Fund) sustain for 20 years if the investments within the RRIF earn 3.6% compounded monthly?

Question 72

In an insurance settlement for bodily injury, a court awarded Mr. Goodman $103,600 for two years' loss of wages of $4,000 per month plus interest on the lost wages to the end of the two years. What effective rate of interest has the court allowed on the lost wages?

Question 73

After contributing $2,000 at the end of each quarter for 13¾ years, Foster has accumulated $205,064 in his RRSP. What effective annual rate of return was earned by the RRSP over the entire period?

Question 74

What maximum annual withdrawals will a $200,000 fund earning 5% compounded annually sustain for 20 years if the withdrawals are made: a) At the beginning of each year b) At the end of each year

Question 75

For $100,000, Royal Life Insurance Co. will sell a 20-year annuity paying $802.76 at the end of each month. What monthly compounded nominal rate and effective rate of return does the annuitant (the buyer of the annuity) earn on the invested funds?

Question 76

Vijay purchased a Government of Nova Scotia bond for $1,050. The bond will pay $35 interest to Vijay at the end of every six months until it matures in seven years. On the maturity date the bond will pay back its $1,000 face value (as well as the interest payment due on that date.) What semi-annually compounded rate of return will Vijay earn during the seven years?

Question 77

What semi-annually compounded rate and effective rate of interest are being charged on a $12,000 loan if semi-annual payments of $1204.55 will repay the loan in seven years?

Question 78

A $9,000, four-year term loan requires monthly payments of $220.77. What are the monthly compounded nominal rate and the effective rate of interest on the loan?

Question 79

What quarterly compounded nominal rate and effective rate of interest are being charged on a $5,000 loan if quarterly payments of $302.07 will repay the loan in 5½ years?

Question 80

A finance company paid a furniture retailer $1934 for a conditional sale contract requiring 12 end-of-month payments of $175. What effective rate of return does the finance company earn on the purchase?