Download pdf - Lloyd M. Clarke - Homevegalee.weebly.com/uploads/2/3/6/...7_4_annuities.pdf · the number of compounding periods per year is 52 ÷ 2, or 26. i 5 0.065__ 26 5 0.0025 n 5 2 26 5 52

444 MHR • Functions 11 • Chapter 7

AnnuitiesHow would you like to be a millionaire without working all your life to earn it? Perhaps if you were lucky enough to win a lottery or have an amazing run on a television game show, it would happen. For most people, however, fantasies such as these are not likely to come true. However, what if you saved money from a very early age? Is it possible to accumulate a million dollars in your lifetime?

When people invest, they usually do not simply deposit one lump sum and wait several years for it to earn interest. Most wise investors makeregularpayments, often deducted directly from their paycheques. Investments of this type are called annuities. In this chapter, you will encounter only ordinarysimpleannuities.

Investigate

Howcanyoudeterminetheamountofanannuity?

Colin is awarded $500 per year as long as he maintains a certain average mark for each of his 4 years of high school. Colin plans to deposit his award at the end of each school year into an account that pays 4% per year, compounded annually.

1. a) Assuming that Colin maintains the necessary average, how many times will he receive $500?

b) Will each of these $500 deposits earn the same interest, in dollars? Explain.

2. a) Determine a method of calculating the total amount of Colin’s investment at graduation.

b) Carry out the method. How much will be in Colin’s account when he graduates?

3. Reflect Consider the method that you used to solve this problem. Would your method be efficient if the number of regular payments was very large? Why or why not?

7.4

regularpayments• payments of equal

value made at equal time periods

annuity• a sum of money paid

as a series of regular payments

ordinaryannuity• an annuity for which

the payments are made at the end of each payment period

simpleannuity• an annuity for which

the compounding and payment periods are the same

Functions 11 CH07.indd 444 6/10/09 4:24:15 PM

Situations like the one described in the Investigate can be represented using a timeline.

The time line shows that a regular payment, R, in dollars, is deposited into an account at the end of each compounding period, for n periods. Because these deposits are made at different times, they will each earn different amounts of interest. For example, the last payment will earn no interest, because it will be received at the end of the annuity. To determine the amount that each of the other deposits will earn, apply the compound interest formula A 5 P(1 i)nto each payment individually.

The amount, A, of the annuity can be determined by adding the amounts of all the payments.

A 5 R R(1 i) R(1 i)2 ... R(1 i)n 2 R(1 i)n 1

Since this is a geometric series with first term a 5 R and common ratio r 5 1 i, use the formula for the sum of a geometric series.

Sn 5 a(rn 1)

__ r 1

5 R[(1 i)n 1]

___ (1 i) 1

5 R[(1 i)n 1]

___ i

The total amount, A, at the time of the last payment of an annuity

can be determined using the formula A 5 R[(1 i)n 1]

___ i , where

R represents the regular payment, in dollars; i represents the interest rate per compounding period, as a decimal; and n represents the number of compounding periods.

This equation can also be written in terms of future value as

FV 5 R[(1 i)n 1]

___ i .

7.4 Annuities • MHR 445

timeline• a diagram used to

illustrate the cash flow of an annuity

R (1 + i )1

R (1 + i )2

R (1 + i )n — 2

R (1 + i )n — 1

Now

R

1

R

2

R

n — 2

R

n — 1

R

n…

…

Compounding Period

Connections

You studied geometric series in Chapter 6 Discrete Functions.

Connections

In this course, you will study only ordinary simple annuities. You will study more complex annuities with, for example, different compounding and payment periods, if you choose to study business at university or college.



Example 1

AmountofanAnnuity

At the end of every month, Hoshi deposits $100 in an account that pays 6% per year, compounded monthly. He does this for 3 years.

a) Draw a time line to represent this annuity.

b) Determine the amount in the account after 3 years.

c) How much interest will the annuity have earned?

Solution

a) Determine the interest per compounding period and the number of compounding periods before drawing the time line.

i 5 0.06 _ 12

5 0.005 n 5 3 12 5 36 R 5 100

b) Method1:UseaScientificCalculator

Substitute the known values into the formula for the amount of an annuity and evaluate.

A 5 R[(1 i)n 1]

___ i

5 100[(1 0.005)36 1]

_____ 0.005

5 100(1.00536 1)

____ 0.005

100 ( 1.005 y x 36 1 ) 0.005 5

3933.61

The amount in Hoshi’s account after 3 years will be $3933.61.

100 (1.005)1

100 (1.005)2

100 (1.005)35

Now

100

1

100

34

100

35

100

36…

…

Time (months)



Method2:UseaTVMSolver

Access the TVM Solver on a graphing calculator and enter the values, as shown.

Move the cursor to the FV field and press ALPHA [SOLVE].

The future value of the sum of Hoshi’s payments is $3933.61. This is the amount in his account after 3 years.

c) To determine the interest earned, calculate the difference between the actual dollar sum of Hoshi’s payments and the future value of the annuity. Hoshi made 36 payments of $100, for a total of $3600.

Interest 5 3933.61 3600 5 333.61 Hoshi’s annuity earned $333.61 in interest.

Example 2

DeterminetheRegularPayment

Sadia needs $4000 for university tuition when she graduates in 2 years. She plans to make deposits into an account that earns 6.5% per year, compounded bi-weekly.


b) How much should she deposit bi-weekly?

Solution

a) Bi-weekly means every 2 weeks. Since there are 52 weeks in a year, the number of compounding periods per year is 52 ÷ 2, or 26.

i 5 0.065 __ 26

5 0.0025

n 5 2 26 5 52

When solving for an annuity, enter the number of payments for N.

The negative sign indicates that payments are being paid, not received.

Both the number of payments and the compounding periods per year are 12.



The regular payment, R, is unknown, but the total future value of the annuity, A, is $4000. A time line representing this annuity is shown.

b) Use the formula for the amount of an annuity to solve for the regular payment, R.

Method1:SubstituteandThenRearrange

A 5 R[(1 i)n 1]

___ i

4000 5 R[(1 0.0025)52 1]

____ 0.0025

10 5 R(1.002552 1)

R 5 10 ___ 1.002552 1

R 72.13

Sadia should deposit $72.13 bi-weekly to have $4000 in 2 years.

Method2:RearrangeandThenSubstitute

A 5 R[(1 i)n 1]

___ i

Ai 5 R[(1 i)n 1]

R 5 Ai___ (1 i)n 1

R 5 4000(0.0025)

____ (1 0.0025)52 1

R 72.13

Sadia should deposit $72.13 every 2 weeks to have $4000 in 2 years.

R (1.0025)1

R (1.0025)2

R (1.0025)51

A = 4000

Now

R

1

R

50

R

51

R

52…

…

Time (2-week periods)

Substitute the known values.

Multiply both sides by 0.0025.

Divide both sides by 1.002552 — 1.

Multiply both sides by i.

Divide both sides by (1 + i )n — 1.

Substitute the known values.



Example 3

DeterminetheInterestRate

Oliver plans to invest $2000 quarterly for 5 years. His financial advisor informs him that his investment will grow to $45 682.40 after the 5 years. What annual rate of interest, compounded quarterly, is Oliver’s annuity earning?

Solution

List the known information for this ordinary simple annuity.

A 5 45 682.40

R 5 2000

n 5 5 4

5 20

Substitute the known values into the equation for the amount of an annuity and solve for i.

45 682.40 5 2000[(1 i)20 1]

____ i

This equation is difficult to solve using standard algebraic techniques. A method of systematic trial could be used, but it would be time-consuming.

Method1:ApplyGraphicalAnalysis

Graph each side of the equation 45 682.40 5 2000[(1 i)20 1]

____ i as a

separate function using a graphing calculator, and use the Intersect operation.

The point of intersection is (0.013 75, 45 682.4), which means that the account will grow to $45 682.40 at the end of the annuity when the interest rate per compounding period is 0.013 75, or 1.375%. To determine the annual interest rate, multiply by the number of compounding periods per year.

1.375% 4 5 5.5%

Oliver’s annuity is earning interest at a rate of 5.5%, compounded quarterly.

Payments are made 4 times a year for 5 years.



Method2:UseaTVMSolver

Access the TVM Solver on a graphing calculator and enter the values, as shown.

Move the cursor to the I% field and pressALPHA [SOLVE].

Recall that I% represents the annual interest rate, as a percent. Oliver’s annuity is earning interest at a rate of 5.5%, compounded quarterly.

Method3:UseaTI-Nspire™CASGraphingCalculator

Use the solve function of the TI-Nspire™ CAS graphing calculator to solve for the interest rate.

• From the home screen, select 6:NewDocument.Then, select 1:AddCalculator.

• Press b. Select 3:Algebra, and then select 1:Solve.

• Type the equation, and then press , I ·.

The interest rate per compounding period is 0.013 75, or 1.375%. To determine the annual interest rate, multiply by the number of compounding periods per year.

1.375% 4 5 5.5%

Therefore, Oliver’s annuity is earning interest at a rate of 5.5%, compounded quarterly.

Example 4

VarytheConditionsofanAnnuity

Felicia plans to invest $2600 at 6% per year, compounded annually, for the next 15 years. Compare the effects on the final amount if the deposits are made and compounding periods are

• annual

• quarterly

• monthly

• weekly



Solution

Method1:UseaGraphingCalculator

Use a table to organize the values of the variables needed to use the

formula for the amount of an annuity A 5 R[(1 i)n 1]

___ i . Note that the

regular payment, R, must be divided by the number of payments per year in each scenario.

Compounding Period R i n

annual 2600 0.06 15

quarterly 650 0.015 60

monthly 216.67 0.005 180

weekly 50 0.001 154 780

Calculate the amount for the first scenario, annual compounding, using a graphing calculator.

Felicia will have $60 517.52 at the end of 15 years if interest is compounded annually.

Repeat the calculation for the other scenarios.

• Press 2nd [ENTRY] to recall the previous calculation.

• Use the cursor keys, the DEL key, and the INS key to modify the equation to fit each scenario.

• Press ENTER to perform the new calculation.

The results of these calculations are summarized in the table.

Compounding Period

Amount($)

annual 60 517.52

quarterly 62 539.52

monthly 63 011.69

weekly 63 198.53

The amount of the annuity increases as the compounding interval becomes more frequent. The difference between weekly and annual compounding is $63 198.53 $60 517.52, or $2681.01.

Technology Tip

Press CLEARto start each calculation with a new window screen.



Method2:UseaSpreadsheet

Set up a table in a spreadsheet.

• Type the headings C/Y (compounding periods per year), n (number of compounding periods or payments), i (interest rate per compounding period), R (regular payment), and A (amount).

• Enter the values 1, 4, 12, and 52 in the C/Y column, to represent the number of compounding periods in each scenario.

• Enter the formulas, starting in cell B2 and working to the right: B2 5A2*15 C2 50.06/A2 D2 52600/A2 E2 5(D2*((1C2)^B2-1))/C2

• Use FillDown to evaluate the remaining calculations.

The results of these calculations are summarized in the table.

Compounding Period

Amount($)

annual 60 517.52

quarterly 62 539.52

monthly 63 010.72

weekly 63 194.18

The amount of the annuity increases as the compounding interval becomes more frequent. The difference between weekly and annual compounding is $63 194.18 $60 517.52, or $2676.66.

n = 15 number of payments per year

i = 0.06 ÷ number of payments per year

R = $2600 ÷ number of payments per year

Calculate the amount using the formula

A = R[(1 + i )n — 1

____ i .

Technology TipTo enter a mathematical expression in a cell in Microsoft® Excel, click on the equal sign and type the expression.

Connections

Some of the amounts are slightly different in Method 1 and Method 2. This is due to rounding in the calculation entry steps. Discrepancies such as this become increasingly important when large sums of money are involved. In general, rounding errors can be avoided or minimized by leaving rational values in fraction form or carrying additional decimal places in the calculation process.

KeyConcepts

An annuity is an investment in which regular payments are deposited into an account.

An ordinary simple annuity is one in which payments are made at the end of every payment period and interest is compounded at the end of the same payment period.

The amount, A, of an annuity can be calculated using the formula A 5 R[(1 i)n 1]

___ i ,

where R represents the regular payment; i represents the interest rate per compounding period, as a decimal; and n represents the number of compounding periods.


CommunicateYourUnderstanding

C1 The time line shows an annuity with an annual interest rate of 8%.

a)How often is interest compounded? How can you tell?

b) What is the duration of the annuity? How can you tell?

c)Explain why this annuity can be represented as a geometric series.

d)Identify the first term, a, and the common ratio, r, of the geometric series.

C2 The graphing calculator screen of a solution using the TVM Solver is shown.

a)What is the duration of the annuity? How can you tell?

b)Why is the payment value negative?

c)Why is the future value positive?

400(1.04)1

400(1.04)2

400(1.04)3

400(1.04)4

400(1.04)5

Now

400

1

400

2

400

3

400

4

400

5

400

6

Compounding Periods

A Practise

Forhelpwithquestions1to3,refertoExample1.

1. Calculate the amount of the annuity shown in the time line.

2. To help her granddaughter with university costs, Sasha’s grandmother puts $250 into an account that earns 4.5% per year, compounded annually, at the end of every year for 6 years.


b) Determine the amount of the annuity.

c) How much interest was earned?

3. At the end of every week, for 2 years, Carlo puts $35 into an account that earns 5.2% per year, compounded weekly.


b) Determine the amount of the annuity.

c) How much interest was earned?

Forhelpwithquestions4and5,refertoExample2.

4. How much must be invested at the end of each year, for 4 years, to achieve an amount of $10 000, if interest is earned at a rate of 6.25% per year, compounded annually?

5. Lucy wants to have $18 000 in her account 3 years from now to buy a car. How much must she invest per month, if her account earns 7.2% annual interest, compounded monthly?

400(1.04)1

400(1.04)2

400(1.04)3

400(1.04)4

400(1.04)5

Now

400

1

400

2

400

3

400

4

400

5

400

6

Compounding Periods

Connecting

Problem Solving

Reasoning and Proving

Reflecting

Selecting ToolsRepresenting

Communicating




B ConnectandApply

Forhelpwithquestions6and7,refertoExample3.

6. Donna invests $75 every 2 weeks in an account that earns compound interest bi-weekly. If she does this for 7 years, she will end up with $16 939.83 in the account.

a) How much total interest will have been earned?

b) Determine the annual rate of interest, compounded bi-weekly.

7. Refer to question 6. By how much must the interest rate be increased for the amount to grow to $18 000 after 7 years?

Forhelpwithquestions8to10,refertoExample4.Usethefollowinginformation.Mauriceisplanningtodeposit$160permonthintoanaccountthatearns4.8%annualinterest,compoundedmonthly,for15years.

8. a) Determine the amount in the account at the end of this annuity.

b) How much interest will have been earned?

9. Maurice’s financial advisor suggests that he would improve the value of his annuity if he changed his payments to $40 per week, at the same interest rate, compounded weekly. Do you agree or disagree with the financial advisor? Justify your response with mathematical reasoning.

10. A competing bank offers Maurice 5% per annum, compounded monthly, for his monthly deposits of $160. Which option should Maurice choose? Justify your answer with mathematical reasoning.

• Stick with his current arrangement.

• Follow his financial advisor’s suggestion about increasing the frequency of his deposits.

• Switch to the competing bank.

11. Lee would like to retire at age 60 and is considering two investment options:

Option A: Invest $500 per month beginning at age 20.

Option B: Invest $1000 per month beginning at age 40.

In both cases, the interest is 6% per annum, compounded monthly. Which option pays more interest, and by how much?

12. Pinder wants to be a millionaire before he retires. He plans to save a certain amount every week for 40 years.

a) If he puts money in an investment that earns 7% annual interest, compounded weekly, what amount must Pinder deposit weekly?

b) What other strategies could Pinder use to achieve his goal? Discuss any assumptions you must make.

C Extend13. UseTechnology Use a graphing calculator

or graphing software.

a) Graph the function A 5 100(1.05n 1)

___ 0.05

.

Describe the shape of the graph.

b) Interpret this function, assuming that it is related to the amount of an annuity.

c) Assuming that interest is compounded annually, identify the regular payment and the annual interest rate.

d) Pose and solve two problems related to this function.

Connecting

Problem Solving

Reasoning and Proving

Reflecting

Selecting ToolsRepresenting

Communicating


14. In question 13, the function

A 5 100(1.05n 1)

___ 0.05

represents the amount

of an annuity.

a) Write a function to describe the total principal invested after n compounding periods.

b) Graph the amount and the principal functions on the same set of axes. Describe the shape of the principal function.

c) Describe how the graphs of these two functions represent the interest earned over time. Write a function to represent the interest earned after n compounding periods.

15. MathContest Bethany starts investing $300 per month at 6% per annum, compounded monthly, for 5 years. After 3 years, the interest increases to 9% per annum, compounded monthly. Determine the amount of her investment after 5 years.

A $19 657.37 B $21 975.21

C $21 453.45 D $3975.21

16. MathContest In ABC, a 5 10 mm, b 5 26 mm, and c 5 24 mm. If D is the midpoint of AC and BC is extended to E such that DE 5 24 mm, determine the measure of CED without using a calculator.

A 30° B 60° C 45° D 67°

17. MathContest Given that a2 (a b)2 5 100 and a and b are whole numbers, determine all possible ordered pairs (a, b) that solve this equation.

18. MathContest Given x2 y2 5 2311, 2311 5 (2)(3)(5)(7)(11) 1, and x y 0, where x and y are integers, which of the following is true?

A only x is divisible by 11

B only y is divisible by 11

C both are divisible by 11

D neither is divisible by 11


Felicity is an investment trader for a large firm in Toronto. She uses her company’s money to buy and sell stocks, bonds, and shares to make a profit. Because the stock market is so volatile, Felicity must always have access to the newest information from around the world. Success in her job depends on the use of computers that can perform fast mathematical calculations. Waiting too long to make a trade could cost her company a lot of money. Felicity trained for her career by taking a 4-year bachelor of administrative studies degree at York University.

Career Connection
