Deck 12: Annuities Due

Question 1

Give three examples of an annuity due.

Answer 1

Insurance premiums, rent payments, lease payments, newspaper and magazine subscriptions, membership dues.

Question 2

For the future value of an annuity due, where is the focal date located relative to the final payment?

Answer 2

The focal date is at the end of the last payment interval. Since payments are at the beginning of each payment interval, the focal date is one payment interval after the final payment.

Question 3

Other things being equal, why is the future value of an annuity due larger than the future value of an ordinary annuity?

Answer 3

Each payment in an annuity due earns interest for one more payment interval than the corresponding in an ordinary annuity.

Question 4

Annual contributions of $1000 will be made to a TFSA for 25 years. The contributor expects investments within the plan to earn 7% compounded annually. What will the TFSA be worth after 25 years if the contributions are made:
a. At the end of each year? b. At the beginning of each year?
c. By what percentage does the answer to Part (b) exceed the answer to Part (a)?

Question 5

Quarterly contributions of $1000 will be made to an RESP for 15 years. Assuming that the investments within the plan grow at 8% compounded quarterly, how much will the TFSA be worth after 15 years if the contributions are made:
a. At the end of each quarter? b. At the beginning of each quarter?
c. By what percentage does the answer to Part (b) exceed the answer to Part (a)?

Question 6

What is the future value of $100 invested at the beginning of every month for 25 years if the investments earn:
a) 6% compounded monthly?
b) 8% compounded monthly?

Question 7

Svetlana intends to invest $1000 at the beginning of every six months. If the investments earn 7% compounded semiannually, what will her investments be worth (rounded to the nearest dollar) after:
a) 25 years?
b) 30 years?

Question 8

Your client plans to invest $10,000 at the beginning of each year for the next 14 years. If the invested funds earn 9.1% compounded annually, what will be the total accumulated value after 14 years? (Taken from CIFP course materials.)

Question 9

Your client has systematically contributed $3000 to her RRSP at the beginning of every three months for the past 17 years. If the RRSP has earned 8.8% compounded quarterly, what is its value today? (Taken from CIFP course materials.)

Question 10

Today Gus is making his first annual contribution of $2500 to a TFSA. How much will the plan be worth 16 years from now if it earns 5.25% compounded monthly?

Question 11

Astrid has just opened an RESP for her children with her first quarterly deposit of $1700. What will the RESP be worth 11½ years from now if the investments within the plan earn 7.5% compounded semiannually?

Question 12

Salvatore will contribute $500 to a mutual fund at the beginning of each calendar quarter.
a) What will be the value of his mutual fund after 6½ years if the fund earns 7.6% compounded annually?
b) How much of this amount represents investment earnings?

Question 13

Monarch Distributing Ltd. plans to accumulate funds for the purchase of a larger warehouse seven years from now. If Monarch contributes $10,000 at the beginning of each month to an investment account earning 4.5% compounded semiannually, what amount (rounded to the nearest dollar) will Monarch accumulate by the end of the seven years?

Question 14

If Hans contributes $1500 to his RRSP on February 1, 1990, and every six months thereafter to and including February 1, 2017, what amount will he accumulate in the RRSP by August 1, 2017? Assume that the RRSP will earn 8.5% compounded semiannually. How much of the total will be interest?

Question 15

Many people make their annual RRSP contribution for a taxation year close to the end of the year. Financial advisers encourage clients to contribute as early in the year as possible. How much more will there be in an RRSP at the end of 25 years if annual contributions of $5000 are made at the beginning of each year instead of at the end? Assume that the RRSP will earn:
a) 8% compounded annually
b) 8% compounded monthly

Question 16

For the past 25 years, Giorgio has contributed $2000 to his RRSP at the beginning of every six months. The plan earned 8% compounded annually for the first 11 years and 7% compounded semiannually for the subsequent 14 years. What is the value of his RRSP today?

Question 17

Keiko has already accumulated $150,000 in her RRSP. She intends to continue to grow her RRSP by making contributions of $500 at the beginning of every month. How much will her RRSP be worth 15 years from now if the RRSP earns 8% compounded annually?

Question 18

Johan recently received his annual performance bonus from his employer. He has set up an investment savings plan to which he will contribute $2000 each year from his bonus and $400 per month from his regular salary. Johan will make his initial contributions of $2000 and $400 today. Rounded to the nearest dollar.
a) what will the plan be worth after 25 years if it earns 7.5% compounded monthly?
b) how much did Johan's contributions earn during the 25 years?

Question 19

What will be the amount in an RRSP after 25 years if contributions of $2000 are made at the beginning of each year for the first 10 years and contributions of $4000 are made at the beginning of each year for the subsequent 15 years? Assume that the RRSP will earn 8% compounded quarterly.

Question 20

Fay contributed $3000 per year to her RRSP on every birthday from age 21 to 30 inclusive. She then ceased employment to raise a family and made no further contributions. Her husband Fred contributed $3000 per year to his RRSP on every birthday from age 31 to 64 inclusive. Assuming that both plans earn 8% compounded annually over the years, calculate and compare the amounts in their RRSPs at age 65.

Question 21

Using the Future Value (Due) Chart. An interactive Future Value (Due) Chart is available in this textbook's Online Learning Centre (OLC). Go to the OLC's Student Edition. In the navigation ar, select "Chapter 12" in the drop-down box. In the list of resources for Chapter 12, select "Links in Textbook" and then click on the link named "Future Value (Due) Chart". Use this chart to solve the following problems (rounded to the nearest dollar).
Your client plans to invest $10,000 at the beginning of each year for the next 14 years. If the invested funds earn 9.1% compounded annually, what will be the total accumulated value after 14 years? (Taken from CIFP course materials.)

Question 22

Today Gus is making his first annual contribution of $2500 to a TFSA. How much will the plan be worth 16 years from now if it earns 5.25% compounded monthly?

Question 23

Salvatore will contribute $500 to a mutual fund at the beginning of each calendar quarter.
a) What will be the value of his mutual fund after 6½ years if the fund earns 7.6% compounded annually?
b) How much of this amount represents investment earnings?

Question 24

Keiko has already accumulated $150,000 in her RRSP. She intends to continue to grow her RRSP by making contributions of $500 at the beginning of every month. How much will her RRSP be worth 15 years from now if the RRSP earns 8% compounded annually?

Question 25

Components of the Future Value of an Annuity Due We will use the Fidelity Investments' Growth Calculator referred to in the NET @ssets feature earlier in this section. Go to the Student Edition of this student textbook's Online Learning Centre (OLC). In the navigation bar, select "Chapter 12" in the dropdown box. In the list of resources for Chapter 12, select "Links in Student textbook" and then click on the link named "Growth Calculator."
The bars in the Growth Calculator chart represent future values at the ends of successive years for the annuity specified by the data entered in cells below the chart. Values can be entered for "Years of Investing" (the annuity's term), the "Initial Balance" (initial lump investment), the "Annual Investment" (amount contributed at the beginning of each year), "Rate of Return" (compounded annually), "Inflation Rate" (compounded annually), and "Tax Rate" (applicable to each year's earnings). You can change a variable's value either by manually entering a new value or by dragging the slider located below the cell. The bar chart immediately adjusts to show the effect of the change. Each bar in the chart shows three components of the future value. The chart refers to the three components as Amount Invested, Simple Earnings, and Compound Earnings. The Amount Invested is just the sum of the investments with no earnings. The Simple Earnings represents interest earned on invested capital on a simple-interest basis. Compound Earnings represents interest earned on previously converted interest. The values of these components at the end of the annuity appear in boxes near the bottom of the window. The interactive chart and calculator will adjust for inflation in two respects. If you intend to increase your annual contributions to keep pace with the rate of inflation, check the "Increase annual investment with inflation" box located below the sliders. (This is the type of scenario we will discuss in Section 13.2). For fixed annual investments, delete the check mark from this box. The second optional inflation adjustment is to display the future value (and its components) in constant purchasing power dollars. You can activate this feature by selecting the "Real (Net of Inflation) Dollars" button.
The calculator will also display results on a before-income-tax basis (by setting Tax Rate _ 0%) or on an after-tax basis. The number you enter for Tax Rate is the percentage of each year's investment earnings that will be paid in income tax. For example, if you enter 40% for Tax Rate and 10% for Rate of Return, the after-tax rate of growth will be only 10% _ 0.4(10%) _ 6%. Answer the following questions for Initial Balance _ $0, Rate of Return _ 10%, and Annual Investment _ $1000. Unless otherwise indicated, set Tax Rate _ 0% (to simulate growth within an RRSP) and Inflation Rate _ 0% (for fixed payments and nominal dollar outcomes).
a) How long does it take for the Compound Earnings component to exceed the Simple Earnings component?
b) After 10 years, what is each component's percentage of the future value?
c) After 25 years, what is each component's percentage of the future value?
d) If the annual rate of inflation is 2%, what is the future value after 25 years in constant purchasing power (real) dollars?
e) If the investments are held otside an RRSP and the annual earnings are taxed at 30%, how much less will you have (in nominal dollars) after 25 years than in Part (c)?

Question 26

Using the Cool Million (Due) Chart This chart is the annuity due version of the (ordinary annuity) Cool Million chart described in Problem 50 of Exercise 11.1. Go to the Student Edition on this student textbook's OLC. In the navigation bar, select "Chapter 12" in the drop-down box. In the list of resources for Chapter 12, select "Links in Student textbook" and then click on the link named "Cool Million (Due)." Return to Problem 50 in Exercise 11.1 to review the features of the chart. Use the same initial planning assumptions in both the Cool Million chart and the Cool Million (Due) chart. How much sooner will you become a millionaire if your regular savings are invested at the beginning of each month instead of at the end of each month?

Question 27

For the present value of an annuity due, where is the focal date located relative to the first payment?

Question 28

Other things being equal, why is the present value of an annuity due larger than the present value of an ordinary annuity?

Question 29

If the periodic interest rate for a payment interval is 3%, by what percentage will PV(due) exceed PV?

Question 30

Other factors being equal, is the PV of an annuity due larger if the given nominal discount rate is compounded monthly instead of annually? Explain briefly.

Question 31

An annuity consists of quarterly payments of $950 for 8 years and 9 months. Discounting at 8% compounded quarterly, determine the present value of the annuity if the payments are made:
a. At the end of each quarter?
b. At the beginning of each quarter?
c. By what percentage does the answer to Part (b) exceed the answer to Part (a)?

Question 32

Using a discount rate of 6% compounded monthly, calculate the present value of monthly payments of $325 for 7¼ years if the payments are made:
a. At the end of each month?
b. At the beginning of each month?
c. By what percentage does the answer to Part (b) exceed the answer to Part (a)?

Question 33

What is the present value of an annuity due consisting of semiannual payments of $1000 for 25 years, if money can earn:
a) 6% compounded semiannually?
b) 8% compounded semiannually?

Question 34

Money can earn 6% compounded monthly. What is the present value of beginning-of-month payments of $100 if the payments continue for:
a) 25 years?
b) 30 years?

Question 35

On the date of its financial statements, a company has 4½ years remaining on the lease of a truck. The lease requires payments of $3000 at the beginning of every six months. What book value is reported for the lease liability if the company pays 8% compounded monthly on its medium-term debt?

Question 36

If money can earn 5.25% compounded monthly, what is the value of an annuity consisting of annual payments of $2500 continuing for 16 years? The first payment will be received today.

Question 37

Carmella purchased a refrigerator under a conditional sale contract that required 30 monthly payments of $60.26 with the first payment due on the purchase date. The interest rate on the outstanding balance was 18% compounded monthly.
a) What was the purchase price of the refrigerator?
b) How much interest did Carmella pay during the 30 months?

Question 38

Rino has just purchased a five-year term life insurance policy. For his premium payments, Rino can choose either beginning-of-month payments of $38.50 or beginning-of-year payments of $455. In current dollars, how much will Rino save during the five years by choosing the lower-cost option? Assume that money can earn 4.8% compounded monthly.

Question 39

Bram must choose between two alternatives for $1,000,000 of life insurance coverage for the next ten years. The premium quoted to him by Sun Life Insurance Co. is $51.75 per month. Atlantic Life will charge $44.25 per month for the first five years and $60.35 per month for the subsequent five years. In both cases, monthly premiums are payable at the beginning of each month. Which policy is "cheaper" if money can earn 4.8% compounded monthly? In current dollars, how much will Bram save by choosing the less costly policy?

Question 40

Under the headline "Local Theatre Project Receives $1 Million!" a newspaper article explained that the Theatre Project had just received the first of 10 annual grants of $100,000 from the Hinton Foundation. What is the current economic value of all of the grants if money is worth 7.5% compounded monthly?

Question 41

You have received two offers on the used car you wish to sell. Mr. Lindberg is offering $8500 cash, and Mrs. Martel's offer is five semiannual payments of $1900, including a payment on the purchase date. Which offer has the greater economic value at a discount rate of 10% compounded semiannually? What is the economic advantage (in current dollars) of the preferred alternative?

Question 42

The life expectancy of the average 65-year-old Canadian male is about 16 additional years. Karsten wants to have sufficient funds in his RRIF at age 65 to be able to withdraw $40,000 at the beginning of each year for the expected survival period of 16 years. If his RRIF earns 6% compounded annually, what amount must he have in the RRIF at the time he turns 65?

Question 43

A rental agreement requires the payment of $900 at the beginning of each month.
a) What single payment at the beginning of the rental year should the landlord accept instead of 12 monthly payments if money is worth 6% compounded monthly?
b) Show that the landlord will be equally well off at the end of the year under either payment arrangement if rental payments are invested at 6% compounded monthly.

Question 44

What minimum amount of money earning 9% compounded semiannually will sustain withdrawals of $1200 at the beginning of every month for 15 years?

Question 45

The lease contract for a computer workstation requires quarterly payments of $2100 at the beginning of every three month period for five years. The lessee would otherwise have to pay an interest rate of 10% compounded quarterly to borrow funds to purchase the workstation.
a) What amount will the lessee initially report in its financial statements as the long-term lease liability?
b) What will the liability be at the end of the fourth year?

Question 46

Beaudoin Haulage has signed a five-year lease with GMAC on a new dump truck. Beaudoin intends to capitalize the lease and report it as a long-term liability. Lease payments of $2700 are made at the beginning of each month. To purchase the truck, Beaudoin would have had to borrow funds at 9% compounded monthly.
a) What initial liability should Beaudoin report on its balance sheet?
b) How much will the liability be reduced during the first year of the lease?

Question 47

What is the current economic value of an annuity due consisting of 22 quarterly payments of $700 if money is worth 6% compounded quarterly for the first three years and 7% compounded quarterly thereafter?

Question 48

Calculate and rank the economic values of the following cash flow streams:
(i) A single payment of $10,000 eight years from now.
(ii) An annuity due starting today with eight annual payments of $850.
(iii) An annuity due starting in eight years with eight annual payments of $1700. Do the calculations and ranking for each of two cases:
a) Money can earn 8% compounded annually for the next 16 years.
b) Money can earn 10% compounded annually for the next 16 years.

Question 49

Two insurance companies gave the following quotations on premiums for essentially the same long-term disability insurance coverage for a 25-year-old. Paul Revere Insurance Co. quoted monthly premiums of $54.83 from ages 26 to 30 inclusive and $78.17 from ages 31 to 64 inclusive. The monthly premiums from Provident Insurance Co. are "flat" at $69.35 from ages 26 to 64 inclusive. All premiums are paid at the beginning of each month. The insurance broker recommended the Provident coverage because the aggregate lifetime premiums up to the client's 65^th birthday are $32,455.80 versus $35,183.16 for the Paul Revere policy. Is the choice that simple? (Hint: Calculate and compare the economic value on the client's 26^th birthday of each policy's stream of premiums assuming a time value of money of 9% compounded monthly.)

Question 50

The lease on the premises occupied by the accounting firm of Heath and Company will soon expire. The current landlord is offering to renew the lease for seven years at $2100 per month. The developers of a new building, a block away from Heath's present offices, are offering the first year of a seven year lease rent-free. For the subsequent six years the rent would be $2500 per month. All rents are paid at the beginning of each month. Other things being equal, which lease should Heath accept if money is worth 7.5% compounded monthly?

Question 51

An ordinary annuity and an annuity due have the same future value, n, and i. Which annuity has the larger payment? Give the reason for your answer.

Question 52

An ordinary annuity and an annuity due have the same present value, n, and i. Which annuity has the smaller payment? Give the reason for your answer.

Question 53

Other variables being the same, how will the amount of the down payment on a lease affect the size of the lease payments?

Question 54

Other variables being the same, how will the size of the residual value affect the size of the car lease payments?

Question 55

The term of the lease on a vehicle is about to expire. Answer parts a and b strictly on financial considerations.
a) If the market value of the vehicle is less than the residual value, what should the lessee do?
b) If the market value of the vehicle exceeds the residual value, what should the lessee do?
c) In view of your answers to a and b, will the interest rate on a lease contract tend to be higher or lower than the interest rate on a loan to purchase the same vehicle? Explain.

Question 56

In order to accumulate $750,000 after 25 years, calculate the amounts that must be invested at the beginning of each year if the invested funds earn:
a. 6% compounded annually.
b. 7% compounded annually.
c. 8% compounded annually.
d. 9% compounded annually.
Also calculate the total earnings in each case.

Question 57

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:
a. 3% compounded monthly?
b. 4.5% compounded monthly?
c. 6% compounded monthly?
d. 7.5% compounded monthly?
In each case, also calculate the total earnings distributed over the life of the annuity.

Question 58

Rounding up the number of contributions to the next integer, how long will it take an RRSP to grow to $600,000 if it receives a contribution of $2500 at the beginning of each quarter and it earns:
a. 6% compounded quarterly?
b. 7% compounded quarterly?
c. 8% compounded quarterly?
d. 9% compounded quarterly?

Question 59

Rounding up to the next month, for how long will a $100,000 fund sustain beginning-of-month withdrawals of $700 if the fund earns:
a. 4% compounded monthly?
b. 5% compounded monthly?
c. 6% compounded monthly?
d. 7% compounded monthly?

Question 60

Ichiro is checking potential outcomes for the growth of his RRSP. He plans to make contributions of $500 at the beginning of each month. What nominal rate of return must his RRSP earn for its future value after 25 years to be:
a. $400,000?
b. $500,000?
c. $600,000?

Question 61

Gina has $500,000 accumulated in her RRSP and intends to use the amount to purchase a 20-year annuity. She is investigating the size of quarterly payment she can expect to receive, depending on the rate of return earned by the funds. What nominal rate of return must the funds earn for the beginning-of-quarter payment to be:
a. $10,000?
b. $11,000?
c. $12,000?

Question 62

Your client currently has accumulated capital of $560,000 and hopes to retire this year. She wants to receive an annuity payment at the beginning of each year for the next 20 years. If the capital can earn
6.5% compounded annually, what maximum annual payment can she receive and just deplete the capital after 20 years? (Taken from CIFP course materials.)

Question 63

Your client purchases an annuity for $700,000 that provides beginning-of-month payments for 15 years. If the annuity earns 4.5% compounded monthly, what monthly payment will he receive? (Taken from CIFP course materials.)

Question 64

Your client wants to accumulate $1,000,000 over the next 25 years by investing the same amount at the beginning of each month. If she can expect a long-term rate of return of 8% compounded annually, how much must she invest each month? (Taken from CIFP course materials.)

Question 65

To accumulate $200,000 after 20 years, what amount must be invested each year if the investment earns
9% compounded annually and the contributions are made:
a) At the beginning of each year?
b) At the end of each year?

Question 66

What maximum annual withdrawals will a $200,000 fund earning 6% 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 67

Triex Manufacturing wants to accumulate $500,000 for an expansion planned to begin in 5 years. If today Triex makes the first of equal quarterly payments into a fund earning 8.25% compounded monthly, what size should these payments be?

Question 68

Your client has already accumulated $20,000 and plans to invest another $5000 at the beginning of each year for the next 15 years. He expects to earn a return of 7 % compounded annually on his investments. How much will his investments be worth fifteen years from now? (Taken from CIFP course materials.)

Question 69

An insurance company wishes to offer customers a monthly instalment alternative to the annual premium plan. All premiums are payable at the beginning of the period of coverage. The monthly payment plan is to include an interest charge of 12% compounded monthly on the unpaid balance of the annual premium. What will be the monthly premium per $100 of annual premium?

Question 70

Advance Leasing calculates the monthly payments on its three-year leases on the basis of recovering the capital cost of the leased equipment and earning an 13.5% compounded monthly rate of return on its capital investment. What will be the monthly lease payment on equipment that costs $8500?

Question 71

Shane is about to have his 25^th birthday. He has set a goal of retiring at age 55 with $700,000 in an RRSP. For planning purposes he is assuming that his RRSP will earn 8% compounded annually.
a) What contribution on each birthday from age 25 to 54 inclusive will be required to accumulate the desired amount in his RRSP?
b) If he waits five years before starting his RRSP, what contribution on each birthday from age 30 to 54 inclusive will be required to accumulate the target amount?

Question 72

Wendy will soon turn 33. She wants to accumulate $500,000 in an RRSP by her 60^th birthday. How much larger will her annual contributions have to be if they are made at the end of each year (from age 33 to age 60) instead of at the beginning of each year? Assume that her RRSP will earn 9% compounded annually.

Question 73

Island Water Taxi has decided to lease another boat for five years rather than to finance the purchase of the boat at an interest rate of 10.5% compounded monthly. It is treating the lease as a capital lease and has set up a long-term lease liability of $43,000. What is the lease payment at the beginning of each month?

Question 74

The MSRP on a Nissan Maxima 3.5 SV is $38,625. The interest rate on a 48-month lease is 1.9% compounded monthly. What is the monthly lease payment, assuming a down payment of $5400 and a residual value of $11,990?

Question 75

The $219.40 monthly payment on a 48-month lease of a Kia SOUL was based on a down payment of $1545, an interest rate of 3.9% compounded monthly, and a residual value of $6815. What is the full price (MSRP) for the car?

Question 76

With a down payment of $4850, the monthly payment on a four-year lease of a Ford F150 SuperCab (MSRP $27,629) is $369.27. The interest rate on the lease is 7.99% compounded monthly. What residual value was used in the calculation?

Question 77

A Smart For Two cabriolet (MSRP $21,550) can be leased for $248 per month. This payment is based on an interest rate of 6.9% compounded monthly, a down payment of $1425, and a residual value of $14,794. What is the term of the lease?

Question 78

What interest rate is being charged if the monthly payment on a 48-month lease of a Jaguar XF (MSRP $58,125) is $799? The required down payment is $2999 and the residual value is $24,059.

Question 79

The MSRP for a BMW 528i is $58,499. The monthly payment on a 48-month lease at 1.9% compounded monthly is $697. The residual value at the end of the lease is $21,000. Rounded to the nearest dollar, what down payment was used in the lease calculation?

Question 80

Kim wants to save half of the $30,000 purchase price of a new car by making monthly deposits of $700, beginning today, into a T-bill savings account earning 4.2% compounded monthly. How long will it take him to reach his goal?