Deck 12: Annuities: Special Situations

Question 1

Marion's grandfather will establish a trust that will pay her $1,500 every three months for 11 years. The first payment will be made six years from now, when she turns 19. If money is worth 6.5% compounded quarterly, what is today's economic value of the bequest?

Answer 1

$32,365.24

Question 2

What is the current economic value of an inheritance that will pay $2,000 to the beneficiary at the beginning of every three months for 20 years, starting when the beneficiary reaches 20 years of age, 4½ years from now? Assume that money is worth 6% compounded monthly. (Round to the nearest dollar.)

Answer 2

$71,795.00

Question 3

Using an inheritance he recently received, Sam wants to purchase a deferred annuity that will pay $5,000 every three months between age 60 (when he plans to retire) and age 65 (when his permanent pension will begin). The first payment is to be three months after he reaches 60, and the last is to be on his 65^th birthday. If Sam's current age is 50 years and 6 months, and the invested funds will earn 4.4% compounded quarterly, what amount must he invest in the deferred annuity?

Answer 3

$58,943.07

Question 4

A $35,000 loan bearing interest at 10% compounded quarterly was repaid, after a period of deferral, by quarterly payments of $1573.83 over 12 years. What was the time interval between the date of the loan and the first payment?

Question 5

A $20,000 investment will be allowed to grow at 4.5% compounded semi-annually until it can support semi-annual withdrawals of $1,000 for 20 years. Rounded to the nearest month, how long before the first withdrawal must the investment be allowed to grow?

Question 6

Nancy borrowed $8,000 from her grandfather to buy a car when she started college. The interest rate being charged is only 4.5% compounded monthly. Nancy is to make the first $200 monthly payment on the loan three years after the date of the loan. How long after the date of the initial loan will she make the final payment?

Question 7

What amount of money invested now will provide payments of $500 at the end of every month for five years following a four-year period of deferral? The money will earn 5.4% compounded monthly.

Question 8

How long is the period deferral if the first quarterly payment of a deferred annuity will be paid 3½ years from today?

Question 9

Mrs. Corriveau has just retired at age 58 with $299,317 in her RRSP. She plans to live off other savings for a few years and allow her RRSP to continue to grow on a tax-deferred basis until there is a sufficient amount to purchase a 25-year annuity paying $2,000 at the end of each month. If her RRSP and the annuity each earn 3.75% compounded monthly, how much longer must she let her RRSP grow (before she buys the annuity)?

Question 10

If money can earn 10% compounded annually for the next 20 years, which of the following annuities has the greater economic value today: $1,000 paid at the end of each of the next 10 years, or 10 annual payments of $2,000 with the first payment occurring 11 years from today?

Question 11

$10,000 was invested in a fund earning 7.5% compounded monthly. How many monthly withdrawals of $300 can be made if the first occurs 3½ years after the date of the initial investment? Count the final smaller withdrawal.

Question 12

What is meant by a deferred annuity?

Question 13

For the same n, PMT, and i, would it cost more or less to purchase a deferred annuity than an ordinary annuity? Explain.

Question 14

Mr. Donatelli moved from Toronto to Winnipeg to take a job promotion. After selling their Toronto home and buying a home in Winnipeg, the Donatellis have $85,000 in cash on hand. If the funds are used to purchase a deferred annuity from a life insurance company providing a rate of return of 8.25% compounded annually, what payments will they receive at the end of every six months for 20 years after a 9-year deferral period?

Question 15

Mr. Haddit plans to retire eight years from today. He projects that he will need $30,000 per year in his retirement which he assumes will be for 15 years. The first payment will be nine years from today. To fund his retirement, Mr. Haddit will invest a lump amount today and later use it to sustain the 15 withdrawals. If his investment earns 6% compounded annually, how much must he invest today?

Question 16

During a one-week promotion, Al's Appliance Warehouse is planning to offer terms of "nothing down and nothing to pay for four months" on major appliances priced above $500. Four months after the date of sale, the first of eight equal monthly payments is due. If the customer is to pay interest at the rate of 12% compounded monthly on the outstanding balance from the date of sale, what will be the monthly payments on an automatic dishwasher priced at $995?

Question 17

What amount of money invested now will provide monthly payments of $200 for five years, if the ordinary annuity is deferred for 3½ years and the money earns 3.75% compounded monthly?

Question 18

Negotiations between Delco Manufacturing and the union representing its employees are at an impasse. The union is seeking a 4.5% wage increase. Delco's offer is 2%. The employees have passed a vote authorizing job action. Suppose the union succeeds in winning the 4.5% increase after a two-month strike. For an employee 10 years from retirement, will there be any economic gain? Compare the current economic values of (1) 10 years' end-of-month wages at the employer's offer (102% of last year's wages) vs. (2) wages including a 4.5% increase to the same horizon but after a two-month strike. Assume money is worth 6% compounded monthly.

Question 19

A conditional sale contract requires the debtor to make six quarterly payments of $569, with the first payment due in six months. What amount will a finance company pay to purchase the contract on the date of sale if the finance company requires a rate of return of 16% compounded quarterly?

Question 20

Why do we terminate the period of deferral one payment interval before the first payment?

Question 21

The common shares of Unicorp. are forecast to pay annual dividends of $2 at the end of each of the next five years, followed by dividends of $3 per year in perpetuity. What is the fair market value of the shares if the market requires a 8% annually compounded rate of return on shares having a similar degree of risk?

Question 22

How much more money is required to fund an ordinary perpetuity than a 25-year ordinary annuity if the funds can earn 7% compounded quarterly and both pay $500 monthly?

Question 23

What amount is required to fund a perpetuity that pays $10,000 at the beginning of each quarter? The funds can be invested to earn 5% compounded quarterly.

Question 24

Ranger Oil recently donated $750,000 to the Northern Alberta Institute of Technology (NAIT) to fund (in perpetuity) five annual bursaries for students in Petroleum Engineering Technology. If the first five bursaries are to be awarded immediately, what is the maximum amount of each bursary? Assume that the bursary fund earns 4.9% compounded semi-annually.

Question 25

Mr. Larsen's will directed that $200,000 be invested to establish a perpetuity making payments at the end of each month to his wife for as long as she lives and subsequently to the Canadian Heart Foundation. What will the payments be if the funds can be invested to earn 5.4% compounded monthly?

Question 26

Mrs. McTavish wants to establish an annual $5,000 scholarship in memory of her husband. The first scholarship is to be awarded two years from now. If the funds can earn 6.25% compounded annually, what amount must Mrs. McTavish pay now to sustain the scholarship in perpetuity?

Question 27

A property development company obtained a $2.5 million loan to construct a commercial building. The interest rate on the loan is 5% compounded semi-annually. The lender granted a period of deferral until rental revenues become established. The first quarterly payment of $100,000 is required 21 months after the date of the loan. How long after the date of the original loan will the last payment be made?

Question 28

A perpetuity is to pay $10,000 at the end of every six months. How much less money is required to fund the perpetuity if the money can be invested to earn 5% compounded semi-annually?

Question 29

What sum of money, invested today in a perpetual fund earning 5.5% compounded semi-annually, will sustain quarterly perpetuity payments of $1,000 if the first payment is made:
a) three months from today?
b) one year from today?

Question 30

The alumni association of Nigara College is initiating a one-year drive to raise money for a perpetual scholarship endowment fund. The goal is to offer ten scholarships per year, each worth $5,000.
a) How large a fund is required to begin awarding the scholarships one year after the funds are in place if the funds can be invested to earn 5% compounded annually in perpetuity?
b) Suppose that, during its fundraising year, the alumni association finds an insurance company that will pay 5.5% compounded annually in perpetuity. How much less money does the association need to raise?
c) What dollar amount in scholarships can be awarded annually if the alumni association raises only $750,000? Use the interest rate from part b.

Question 31

What percentage more money is required to fund an ordinary perpetuity than to fund a 30-year ordinary annuity if the funds can earn 5.8% compounded semi-annually? The perpetuity and the annuity each pay $1,000 semi-annually.

Question 32

Mr. O'Connor set up a trust account paying $500 per month in perpetuity to the local hospital. These payments consume all the interest earned monthly by the trust. Between what amounts does the balance in the trust account fluctuate if it earns 6% compounded monthly?

Question 33

An old agreement requires a town to pay $500 per year in perpetuity to the owner of a parcel of land for a water-well dug on the property in the 1920s. The well is no longer used, and the town wants to buy out the contract, which has become an administrative nuisance. What amount (including the regular scheduled payment) should the landowner be willing to accept on the date of the next scheduled payment if long-term low-risk investments now earn 3.8% compounded annually?

Question 34

If money can earn 6% compounded annually, what percentage more money is required to fund an ordinary perpetuity paying $1,000 at the end of every year than to fund an ordinary annuity paying $1,000 per year for 25 years?

Question 35

A company's preferred shares pay a $2 dividend every six months in perpetuity. What is the fair market value of the shares just after payment of a dividend if the dividend yield required by the market on shares of similar risk is
a) 4% compounded semi-annually?
b) 5% compounded semi-annually?

Question 36

A city sells plots in its cemetery for $1,000 plus an amount calculated to provide for the cost of maintaining the grounds in perpetuity. This cost is figured at $25 per plot due at the end of each quarter. If the city can invest the funds to earn 4.8% compounded annually in perpetuity, what is the price of a plot?

Question 37

A company's preferred shares pay a $1.25 dividend every three months in perpetuity. What is the fair market value of the shares just after payment of a dividend if the rate of return required by the market on shares of similar risk is:
a) 5% compounded quarterly?
b) 6% compounded quarterly?

Question 38

How much more money is required to fund an ordinary perpetuity than a 30-year ordinary annuity if both pay $5,000 quarterly and money can earn 5% compounded quarterly?

Question 39

Dr. Pollard donated $100,000 to the Canadian National Institute for the Blind. The money is to be used to make semi-annual payments in perpetuity (after a period of deferral) to finance the recording of books on tape. The first perpetuity payment is to be made five years from the date of the donation. If the funds are invested at 5% compounded semi-annually, what will be the size of the payments?

Question 40

Mrs. O'Reilly donated $500,000 to Medicine Hat College for a perpetual scholarship fund for women in business studies. What amount can be awarded on each anniversary if the scholarship fund earns 4½% compounded annually?

Question 41

A legal dispute delayed for 18 months the disbursement of a $500,000 bequest designated to provide quarterly payments in perpetuity to a hospice. While under the jurisdiction of the court, the funds earned interest at the rate of 5% compounded semi-annually. The hospice has just invested the $500,000 along with its earnings in a perpetual fund earning 5.2% compounded semi-annually. What payments will the hospice receive beginning three months from now?

Question 42

Dean has already implemented the first stage of his financial plan. Over a 30-year period, he will continue to increase his annual year-end RRSP contributions by 2% per year. His initial contribution was $2,000. At the end of the 30 years, he will transfer the funds to an RRIF and begin end-of-month withdrawals that will increase at the rate of 1.8% compounded monthly for 25 years. Assume that his RRSP will earn 6% compounded annually and his RRIF will earn 3% compounded monthly. What will be the size of his initial RRIF withdrawal?

Question 43

Suppose year-end contributions to an RRSP start at $3,000 and increase by 2.5% per year thereafter. What will be the amount in the RRSP after 25 years if the plan earns 9% compounded annually?

Question 44

Maritime Bank recently announced that its next semi-annual dividend (to be paid six months from now) will be $1.00 per share. A stock analyst's best estimate for the growth in future dividends is 4% compounded semi-annually.
a) If you require a rate of return of 7.8% compounded semi-annually on the stock, what maximum price should you be willing to pay per share? Ignore the present value of dividends beyond a 50-year time horizon.
b) What price do you obtain if you do not ignore dividends beyond 50 years? (Hint: Use a large value, say 1000, for n in the present value calculation.)

Question 45

Mrs. Kirkpatrick (age 65) is about to begin receiving a CPP retirement pension of $9,000 per year. This pension is indexed to the Consumer Price Index (CPI). Assume that the annual pension will be paid in a single year-end payment, the CPI will rise 3% per year, and money is worth 6% compounded annually. What is the current economic value of:
a) 20 years of pension benefits? b) 25 years of pension benefits?

Question 46

The common shares of Bancorp Ltd. are forecast to pay annual dividends of $3 at the end of each of the next five years, followed by dividends of $2 per year in perpetuity. What is the fair market value of the shares if the market requires a 5% annually compounded rate of return on shares having a similar degree of risk?

Question 47

A wealthy benefactor has donated $1,000,000 to establish a perpetuity that will be used to support the operating costs of a local heritage museum scheduled to open in 3 years' time. If the funds earn 4.8% compounded monthly, what monthly payments, the first occurring three years from now, can the museum expect?

Question 48

Randall wants to accumulate $750,000 in his RRSP by the end of his 30-year working career. What should be his initial year-end contribution if he intends to increase the contribution by 3% every year and the RRSP earns 5.25% compounded annually?

Question 49

The dividends on the common shares of Mosco Inc. are forecast to grow at 10% per year for the next five years. Thereafter, the best guess is that the annual dividend will grow at the same 3% annual rate as the nominal GNP. A $2.00 dividend for the past year was recently paid. Assume that the required rate of return is 9% compounded annually. What is the fair market value of the shares if we ignore all dividends beyond a 30-year time horizon?

Question 50

Determine the unknown value for the following deferred annuity. The annuity is understood to be an ordinary annuity after the period of deferral.

Question 51

Chantal will make year-end contributions for 30 years to an RRSP earning 8% compounded annually.
a) How much will she have after 30 years if the annual contribution is $2,000?
b) How much more will she have after 30 years if she increases the contributions by 2% every year?

Question 52

Ken Tuckie is about to buy a 25-year annuity that will deliver end-of-month payments. The first payment will be $1,000. How much more will it cost to index the annuity so that payments grow at the rate of 2.4% compounded monthly? Assume the money used to purchase the annuity earns 5.4% compounded monthly.

Question 53

Mr. Chan has donated $1 million to a college to set up a perpetuity for the purchase of books and journals for a new library to be built and named in his honour. The donation will be invested and earnings will compound for three years, at which time the first of the quarterly perpetuity payments will be made. If the funds earn 6% compounded quarterly, what will be the size of the payments?

Question 54

In this LO, does constant growth mean that each successive payment increases by the same dollar amount? If not, what does it mean?

Question 55

How much will it cost to purchase a 20-year indexed annuity in which the end-of-quarter payments start at $5,000 and grow by 2% compounded quarterly? Assume that the money used to purchase the annuity earns 6% compounded quarterly.

Question 56

Determine the unknown value for the following deferred annuity. The annuity is understood to be an ordinary annuity after the period of deferral.

Question 57

How would you handle cases where successive annuity payments decrease by the same percentage every payment interval?

Question 58

Ida Ho is about to retire from a government job with a pension that is indexed to the Consumer Price Index (CPI). She is 60 years old and has a life expectancy of 25 years. Estimate the current economic value of her pension which will start at $20,000 per year? For the purpose of this estimation, assume that Ida will draw the pension for 25 years, the annual pension will be paid in a single year-end payment, the CPI will rise 2.5% per year, and money is worth 5% compounded annually. How much of the current economic value comes from indexing?

Question 59

What minimum amount will have to be dedicated today to a fund earning 5.6% compounded quarterly if the first quarterly payment of $2,000 in perpetuity is to occur:
a) three months from now?
b) five years from now?

Question 60

Determine the unknown value for the following deferred annuity. The annuity is understood to be an ordinary annuity after the period of deferral.

Question 61

Determine the term, expressed in years and months, for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 62

Determine the periodic payment for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 63

Determine the term, expressed in years and months, for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 64

Determine the term, expressed in years and months, for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 65

Determine the term, expressed in years and months, for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 66

Determine the periodic payment for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 67

Determine the unknown value for the following deferred annuity. The annuity is understood to be an ordinary annuity after the period of deferral.

Question 68

Determine the unknown value for the following deferred annuity. The annuity is understood to be an ordinary annuity after the period of deferral.

Question 69

What is the future value eight years from now of each of the following cash-flow streams if money can earn 9% compounded semi-annually?
a) A single payment of $5,000 today.
b) An ordinary annuity starting today with eight annual payments of $900.
c) An ordinary annuity starting in three years with 20 quarterly payments of $400.

Question 70

Determine the periodic payment for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 71

Determine the term, expressed in years and months, for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 72

Determine the term, expressed in years and months, for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 73

What price will a finance company pay to a merchant for a conditional sale contract that requires 15 monthly payments of $231 beginning in six months? The finance company requires a rate of return of 18% compounded monthly.

Question 74

Determine the unknown value for the following deferred annuity. The annuity is understood to be an ordinary annuity after the period of deferral.

Question 75

Determine the unknown value for the following deferred annuity. The annuity is understood to be an ordinary annuity after the period of deferral.

Question 76

Determine the unknown value for the following deferred annuity. The annuity is understood to be an ordinary annuity after the period of deferral.

Question 77

Determine the periodic payment for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 78

Determine the periodic payment for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 79

Determine the periodic payment for the following deferred annuity. The annuity is an ordinary annuity following the period of deferral.

Question 80

Determine the unknown value for the following deferred annuity. The annuity is understood to be an ordinary annuity after the period of deferral.