Deck 10: Annuities: Future Value and Present Value

Question 1

Markus spends $160 per month on cigarettes. Suppose he quits smoking and invests the same amount at the end of each month for 20 years. If the invested money earns 7.5% compounded monthly, how much will Markus accumulate after 20 years?

Answer 1

$88,596.92

Question 2

The rate of return offered by Reliance Insurance Co. on its 20-year annuities is 4.8% compounded monthly. Thom wishes to purchase an annuity from Reliance that will provide him with month-end payments of $1000 for the next 20 years. How much will Reliance charge Thom for this annuity?

Answer 2

$154,093.30

Question 3

Determine the present value of the ordinary annuity:

Answer 3

$15,502.92

Question 4

Determine the present value of the ordinary annuity:

Question 5

What distinguishes an ordinary simple annuity from an ordinary general annuity?

Question 6

What is meant by the "term" of an annuity?

Question 7

This problem demonstrates the dependence of the future value of an annuity on the interest rate. Suppose $1,000 is invested at the end of each year for 20 years. Calculate the future value if the investments earn an annually compounded rate of return of:
a) 5% b) 6% c) 7% d) 8%
Note that the future value increases proportionately more than the interest rate.

Question 8

The monthly payments on a five-year loan at 7.5% compounded monthly are $200.38.
a) What was the original amount of the loan?
b) What is the balance after the 30^th payment?

Question 9

Determine the present value of the ordinary annuity:

Question 10

Determine the present value of the ordinary annuity:

Question 11

Determine the future value:

Question 12

Determine the present value of the ordinary annuity:

Question 13

This problem demonstrates the dependence of the present value of an annuity on the number of payments. Using 7% compounded annually as the discount rate, calculate the present value of an ordinary annuity paying $1,000 per year for:
a) 5 years b) 10 years c) 20 years d) 30 years e) 100 years f) 1,000 years
Observe that the present value increases with increasing n, but at a diminishing rate. In this case, the 970 payments from Year 30 to Year 1,000 cause the present value to increase by only about 15%.

Question 14

This problem demonstrates the dependence of the future value of an annuity on the number of payments. Suppose $1,000 is invested at the end of each year. Assume the investments earn 5% compounded annually. Calculate the future value of the investments after each of the following numbers of payments:
a) 5 b) 10 c) 15 d) 20 e) 25 f) 30
Note that the future value increases proportionately more than n as n is increased.

Question 15

Determine the future value:

Question 16

Determine the future value:

Question 17

Determine the future value:

Question 18

What is the present value of end-of-quarter payments of $2,500 for seven years? Use a discount rate of 6% compounded quarterly.

Question 19

Determine the present value of payments of $100 at the end of each month for 20 years. Use a discount rate (interest rate) of 6% compounded monthly.

Question 20

Determine the present value of the ordinary annuity:

Question 21

What is the appropriate price to pay for a contract guaranteeing payments of $1,500 at the end of each quarter for the next 12 years? You require a rate of return of 6% compounded quarterly for the first five years and 7% compounded quarterly for the next seven years.

Question 22

Dakota intends to save for occasional major travel holidays by contributing $275 at the end of each month to an investment plan. At the end of every three years, she will withdraw $10,000 for a major trip abroad. If the plan earns 6% compounded monthly, what will be the plan's balance after seven years?

Question 23

You can purchase a residential building lot for $60,000 cash or for $10,000 down and month-end payments of $1,000 for five years. If money is worth 7.5% compounded monthly, which option should you choose?

Question 24

Kent sold his car to Carolynn for $2,000 down and monthly payments of $295.88 for 3½ years, including interest at 7.5% compounded monthly. What was the selling price of the car?

Question 25

Rachelle purchased a motor home for $9,000 down, with the balance to be paid by 60 monthly payments of $1176.40 including interest at 6% compounded monthly.
a) What was the purchase price of the motor home?
b) If the principal balance may be prepaid at any time, what is the payout amount two years after the purchase date (not including the scheduled payment on that date)?

Question 26

Suppose the discount rate used to calculate the present value of an annuity is increased (leaving n and PMT unchanged). Will the annuity's present value be (pick one) (i) larger or (ii) smaller than before? Give the reason for your choice.

Question 27

Annuity G has the same i and PMT as Annuity H. G has twice as many payments as H. Is G's present value (pick one): (i) double, (ii) more than double, or (iii) less than double the amount of H's present value? Give the reason for your choice.

Question 28

Sebastin has made contributions of $2,000 to his RRSP at the end of every six months for the past eight years. The plan has earned 5.5% compounded semi-annually. He has just moved the funds to another plan, paying 4% compounded quarterly. He will now contribute $1,500 at the end of every three months. What total amount will he have in the plan seven years from now?

Question 29

The Ottawa Senators fired their coach two years into his five-year contract, which paid him $90,000 at the end of each month. If the team owners buy out the remaining term of the coach's contract for its economic value at the time of firing, what will be the settlement amount? Use 7.5% compounded monthly as the time value of money.

Question 30

Dr. Krawchuk made deposits of $2,000 to his RRSP at the end of each calendar quarter for 6 years. He then left general practice for specialist training and did not make further contributions for 2½ years. What amount was in his RRSP at the end of this period if the plan earned 5% compounded quarterly over the entire 8½ years?

Question 31

A conditional sale contract between Classic Furniture and the purchaser of a dining room set requires month-end payments of $250 for 15 months. Classic Furniture sold the contract to Household Finance Co. at a discount to yield 19.5% compounded monthly. What price did Household pay Classic Furniture?

Question 32

Charlene has made contributions of $3,000 to her RRSP at the end of every half year for the past seven years. The plan has earned 9% compounded semi-annually. She has just moved the funds to another plan earning 7.5% compounded quarterly, and will now contribute $2,000 at the end of every three months. What total amount will she have in the plan five years from now?

Question 33

Annuity G has the same i and PMT as Annuity H. G has twice as many payments as H. Will G's future value be (pick one): (i) double, (ii) more than double, or (iii) less than double the amount of H's future value? Give the reason for your choice.

Question 34

The Montreal Canadiens have just announced the signing of Finnish hockey sensation Gunnar Skoroften to a 10-year contract at $3 million per year. The media are reporting the deal as being worth $30 million to the young Finn. Rounded to the dollar, what current economic value would you place on the contract if Skoroften will be paid $250,000 at the end of each month and money can earn 6% compounded monthly?

Question 35

Calculate the amounts that will be accumulated after 20 years if:
a) $1,000 is invested at the end of every six months at 8.5% compounded semi-annually.
b) $2,000 is invested at the end of every year at 8.5% compounded annually.

Question 36

Annuity A has the same n and i as Annuity
B. A's PMT is double B's PMT. Will A's future value be (pick one): (i) double, (ii) more than double, or (iii) less than double the amount of B's future value? Give the reason for your choice.

Question 37

Annuity A has the same n and i as Annuity
B. A's PMT is double B's PMT. Will A's present value be (pick one): (i) double, (ii) more than double, or (iii) less than double the amount of B's present value? Give the reason for your choice.

Question 38

A Province of Ontario bond has 14½ years remaining until it matures. The bond pays $231.25 interest at the end of every six months. At maturity, the bond repays its $5,000 face value in addition to the final interest payment. What is the fair market value of the bond if similar provincial bonds are currently providing investors with a return of 3.8% compounded semi-annually?

Question 39

You have received two offers on the used car you wish to sell. Mr. Lindberg is offering $9500 cash, and Mrs. Martel's offer is five semi-annual payments of $2,000 including one on the purchase date. Which offer has the greater economic value using a discount rate of 5% compounded semi-annually? What is the economic advantage in current dollars of the preferred alternative?

Question 40

Think of a 20-year annuity paying $2,000 per month. If prevailing market rates decline over the next year, will the price to purchase a 20-year annuity increase or decrease? Explain.

Question 41

A Government of Canada bond will pay $50 at the end of every six months for the next 15 years and an additional $1,000 lump payment at the end of the 15 years. What is the appropriate price to pay if you require a rate of return of 6.5% compounded semi-annually?

Question 42

Calculate the periodic interest rate that matches the payment interval for each annuity (to the nearest 0.001%):

Question 43

What is the maximum price you should pay for a contract guaranteeing month-end payments of $500 for the next 12 years if you require a minimum rate of return of at least 8% compounded monthly for the first five years and at least 9% compounded monthly for the next seven years?

Question 44

A mortgage broker offers to sell you a loan contract that would provide you with end-of-month payments of $900 for the next years. In addition, at the end of the year period, you will also receive a lump sum payment of $37,886. What should you pay for the contract, if you require a rate of return of 7.2% compounded monthly?

Question 45

Calculate the future value of an ordinary annuity consisting of monthly payments of $300 for five years. The rate of interest was 3% compounded monthly for the first two years and will be 4.5% compounded monthly for the last three years.

Question 46

Determine the future value (accurate to the cent) of the ordinary general annuity:

Question 47

An Agreement for Sale contract on a house requires payments of $4,000 at the end of every six months. The contract has seven years to run. The payee wants to sell her interest in the contract. What will an investor pay in order to realize an annually compounded rate of return on the purchase price of:
a) 8%? b) 10%?

Question 48

Mr. and Mrs. Krenz are contributing to a Registered Education Savings Plan
(RESP) they have set up for their children. What amount will they have in the RESP after eight years of contributing $500 at the end of every calendar quarter if the plan earns 6% compounded monthly? How much of the total amount is interest?

Question 49

An individual qualifying for Canada Pension Plan benefits may elect to start collecting the CPP monthly retirement benefit at any time between the ages of 60 and 70. If the retirement benefit starts after age 65, the pension payments are increased (from the amount that would otherwise be paid at age 65) by 0.5% for each month after age 65. For example, if the retiree chooses to begin receiving the benefit after turning 68, the CPP payments will be increased by (36 months) x (0.5%) = 18%.
The average life expectancy of a man aged 65 is another 15 years. If a man aged 65 lives exactly the expected 15 years, compare the economic values at age 65 of the two alternatives of collecting a 100% pension from age 65 versus a 118% pension from age 68. Assume that money is worth 7.5% compounded monthly.

Question 50

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.

Question 51

Determine the present value (accurate to the cent) of the ordinary general annuity:

Question 52

Kent sold his car to Carolynn for $2,000 down and monthly payments of $295.88 for 3½ years, including interest at 7.5% compounded annually. What was the selling price of the car?

Question 53

A loan at 6% compounded semi-annually requires equal monthly payments. What is the value of c? What is the approximate value of the periodic interest rate for one payment interval? Will the correct value be larger or smaller than your estimate? Explain.

Question 54

An income annuity with quarterly payments earns 6% compounded monthly. What is the value of c? What is the approximate value of the periodic rate of return for one payment interval? Will the correct value be larger or smaller than your estimate? Explain.

Question 55

LeVero's end-of-month payments of $1167.89 will pay off his mortgage loan in 4 years and 7 months. The interest rate on his mortgage is 6.6% compounded semi-annually. What is the current balance on the loan?

Question 56

What are the distinguishing characteristics of an ordinary general annuity?

Question 57

What amount will be required to purchase a 20-year annuity paying $2,500 at the end of each month if the annuity provides a return of 6.75% compounded annually?

Question 58

A court-ordered award for family support calls for payments of $800 per month for five years, followed by payments of $1,000 per month for 10 more years. If money is worth 6% compounded monthly, what is the economic value of the award one month before the first payment?

Question 59

The British Columbia Teachers' Pension Plan allows a teacher to begin collecting a retirement pension before age 60, but the pension is reduced by 3% for each year the retiring teacher's age is under 60. For example, a teacher retiring at age 56 would receive 88% of the monthly pension that she would receive at age 60 (with the same number of years of service). The reduction is permanent, extending to payments beyond age 60.
Suppose that a female teacher will live the average life expectancy of 28 additional years for a woman aged 55. Compare the economic values at age 55 of the two alternatives of collecting an 85% pension from age 55 versus collecting a 100% pension from age 60. Assume that money is worth 7.5% compounded monthly.

Question 60

A 15-year loan requires monthly payments of $587.33 including interest at 8.4% compounded monthly.
a) What was the original amount of the loan?
b) What is the balance on the loan after half of the payments have been made?

Question 61

If an ordinary annuity with quarterly payments and a 5½-year term began June 1, 2010, what are the dates of the first and last payments?

Question 62

Dr. Wilson is buying a 50% partnership in a veterinary practice by end-of- month payments of $714.60, including interest at 7% compounded semi- annually for 15 years.
a) What valuation was placed on the partnership at the beginning of the payments?
b) What total amount of interest will she pay over the 15 years?

Question 63

Determine the future value:

Question 64

A life insurance company quoted an annual premium of $387.50 (payable at the beginning of the year) for a $250,000 term insurance policy on a 35-year-old male nonsmoker. Alternatively, the insured can pay $33.71 at the beginning of each month by preauthorized electronic debit. Which payment plan would an applicant choose solely on the basis of money being worth 3.5% compounded monthly?

Question 65

The Toronto Raptors announce the signing of their top draft pick to a "7-year deal worth $43.2 million." The player will earn $400,000 at the end of each month for the first three years, and $600,000 at the end of each month for the subsequent four years. How do the Raptors get the $43.2 million figure? To the nearest $1,000, what is the true current economic value of the deal if money can earn 7% compounded annually?

Question 66

Gloria has just made her ninth annual $2,000 contribution to her Tax-Free Savings Account (TFSA). She now plans to make semi-annual contributions of $2,000. The first contribution will be made six months from now. How much will she have in her TFSA 15 years from now if the plan has earned and will continue to earn 5% compounded quarterly?

Question 67

Assume that your client invests $1,000 at the end of each of the next three years. The investments earn 8% compounded annually. What is the future value at the end of the three years? (Taken from CIFP course materials.)

Question 68

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 $3600? In both cases the RRSP earns 6.5% compounded semi-annually.

Question 69

Determine the future value:

Question 70

Micheline wishes to purchase a 25-year annuity providing payments of $1,000 per month for the first 15 years and $1,500 per month for the remaining 10 years. Sovereign Insurance Co. has quoted her a rate of return of 5% compounded annually for such an annuity. How much will it cost Micheline to purchase the annuity from Sovereign?

Question 71

Your client has systematically invested $1,000 at the end of each half-year for the past 17 years. The invested funds have earned 6.4% compounded semi-annually. What is the value of your client's investments today? (Taken from CIFP course materials.)

Question 72

Determine the present value of the ordinary annuity:

Question 73

Determine the future value

Question 74

A seven-year capital lease of an executive jet requires semi-annual payments of $200,000 at the beginning of each six-month period. The company can borrow funds for seven years at 7.4% compounded semi-annually.
a. What long-term lease liability will the firm set up at the start of the term of the lease?
b. What liability will remain halfway through the term of the lease

Question 75

Your client plans to invest $2,000 at the end of each year. The rate of return on the investment is 7.5% compounded annually. What will be the value of the investment at the end of 12 years? (Taken from CIFP course materials.)

Question 76

Determine the present value of the ordinary annuity:

Question 77

Determine the future value:

Question 78

How much larger will the value of an RRSP be at the end of 20 years if the contributor makes month-end contributions of $500 instead of year-end contributions of $6,000? In both cases the RRSP earns 4.5% compounded semi-annually.

Question 79

Suppose Evan contributes $2,000 to his RRSP at the end of every quarter for the next 15 years and then contributes $1,000 at each month's end for the subsequent 10 years. How much will he have in his RRSP at the end of the 25 years? Assume that the RRSP earns 6% compounded semi-annually.

Question 80

Calculate the future value of investments consisting of payments of $800 at the end of each calendar quarter for seven years. The rate of interest earned will be 7% compounded quarterly for the first 30 months and 6% compounded semi-annually for the remainder of the annuity's term.