1-1. 1-2 Chapter 13 Annuities and Sinking Funds McGraw-Hill/Irwin Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved

1-1

1-2

Chapter 13Chapter 13

Annuities and Sinking Annuities and Sinking FundsFunds

McGraw-Hill/Irwin Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved.

1-3

• Differentiate between contingent annuities and annuities certain

• Calculate the future value of an ordinary annuity and an annuity due manually and by table lookup

Annuities and Sinking Funds#13#13Learning Unit Objectives

Annuities: Ordinary Annuity and Annuity Due (Find Future Value)

LU13.1LU13.1

1-4

Compounding Interest (Future Value)

Term of the annuity - the time from the beginning of the first payment period to the end of the last payment period.

Future value of annuity - the future dollar amount of a series of payments plus interest

Present value of an annuity - the amount of money needed to invest today in order to receive a stream of payments for a given number of years in the future

Annuity - A series of payments

1-5

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$3.50

1

End of period

$1.00

Figure 13.1 Future value of an annuity of $1 at 8%

1-6

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$3.50

1 2

End of period

$1.00

$2.08


1-7

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$3.50

1 2 3

End of period

$1.00

$2.08

$3.25


1-8

Classification of Annuities

Contingent Annuities - have no fixed number of payments but depend on an uncertain event

Annuities certain - have a specific stated number of payments

Life Insurance payments Mortgage payments

1-9

Classification of Annuities

Ordinary annuity - regular

deposits/payments made at the end of

the period

Annuity due - regular

deposits/payments made at the

beginning of the period

Jan. 31 Monthly Jan. 1

June 30 Quarterly April 1

Dec. 31 Semiannually July 1

Dec. 31 Annually Jan. 1

1-10

Tools for Calculating Compound Interest

Number of periods (N) Number of years times

the number of times the interest is

compounded per year

Rate for each period (R) Annual interest rate

divided by the number of times the interest is

compounded per year

If you compounded $100 each year for 3 years at 6% annually, semiannually, or quarterly What is N and R?

Annually: 3 x 1 = 3Semiannually: 3 x 2 = 6Quarterly: 3 x 4 = 12

Annually: 6% / 1 = 6%Semiannually: 6% / 2 = 3%Quarterly: 6% / 4 = 1.5%

Periods Rate

1-11

Step 1. Calculate the number of periods and rate per period

Step 2. Lookup the periods and rate in an ordinary annuity table. The intersection gives the table factor for the future value of $1

Step 3. Multiply the payment each period by the table factor. This gives the future value of the annuity

Calculating Future Value of an Ordinary Annuity by Table Lookup

1-12

Period 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2 2.0200 2.0300 2.0400 2.0500 2.0600 2.0700 2.0800 2.0900 2.1000

3 3.0604 3.0909 3.1216 3.1525 3.1836 3.2149 3.2464 1.0000 3.3100

4 4.1216 4.1836 4.2465 4.3101 4.3746 4.4399 4.5061 4.5731 4.6410

5 5.2040 5.3091 5.4163 5.5256 5.6371 5.7507 5.8666 5.9847 6.1051

6 6.3081 6.4684 6.6330 6.8019 6.9753 7.1533 7.3359 7.5233 7.7156

7 7.4343 7.6625 7.8983 8.1420 8.3938 8.6540 8.9228 9.2004 9.4872

8 8.5829 8.8923 9.2142 9.5491 9.8975 10.2598 10.6366 11.0285 11.4359

9 9.7546 10.1591 10.5828 11.0265 11.4913 11.9780 12.4876 13.0210 13.5795

10 10.9497 11.4639 12.0061 12.5779 13.1808 13.8164 14.4866 15.1929 15.9374

11 12.1687 12.8078 13.4863 14.2068 14.9716 15.7836 16.6455 17.5603 18.5312

12 13.4120 14.1920 15.0258 15.9171 16.8699 17.8884 18.9771 20.1407 21.3843

13 14.6803 15.6178 16.6268 17.7129 18.8821 20.1406 21.4953 22.9534 24.5227

14 15.9739 17.0863 18.2919 19.5986 21.0150 22.5505 24.2149 26.0192 27.9750

15 17.2934 18.5989 20.0236 21.5785 23.2759 25.1290 27.1521 29.3609 31.7725

Ordinary annuity table: Compound sum of an annuity of $1 (Partial)

Table 13.1 Ordinary annuity table: Compound sum of an annuity of $1

1-13

N = 5 x 1 = 5

R = 9%/1 = 9%

5.9847 x $2,000

$11,969.40

Future Value of an Ordinary Annuity

Find the value of an investment after 5 years for a $2,000 ordinary annuity at 9%

Manual Calculation2,000.00$ End of Yr 1

180.00 9% interest2,180.00

1-14

N = 5 x 1 = 5

R = 9%/1 = 9%

5.9847 x $2,000

$11,969.40




180.00 2,180.00 2,000.00 End of Yr 24,180.00

376.20 9 % interest4,556.20

1-15

N = 5 x 1 = 5

R = 9%/1 = 9%

5.9847 x $2,000

$11,969.40




180.00 2,180.00 2,000.00 End of Yr 24,180.00

376.20 4,556.20 2,000.00 End of Yr 36,556.20

590.06 9% interest7,146.26

1-16

N = 5 x 1 = 5

R = 9%/1 = 9%

5.9847 x $2,000

$11,969.40




180.00 2,180.00 2,000.00 End of Yr 24,180.00

376.20 4,556.20 2,000.00 End of Yr 36,556.20

590.06 7,146.26 2,000.00 End of Yr 49,146.26

823.16 9 % interest9,969.42

1-17

N = 5 x 1 = 5

R = 9%/1 = 9%

5.9847 x $2,000

$11,969.40




180.00 2,180.00 2,000.00 End of Yr 24,180.00

376.20 4,556.20 2,000.00 End of Yr 36,556.20

590.06 7,146.26 2,000.00 End of Yr 49,146.26

823.16 9,969.42 2,000.00 End of Yr 5

11,969.42$

1-18

Calculating Future Value of an Annuity Due by Table Lookup

Step 1. Calculate the number of periods and rate per period. Add one extra period.

Step 2. Look up the periods and rate in an ordinary annuity table. The intersection gives the table factor for the future value of $1

Step 3. Multiply the payment each period by the table factor.

Step 4. Subtract 1 payment from Step 3.

Payment is made at the beginning of the period.

1-19

Future Value of an Annuity Due

Find the value of an investment after 5 years for a $2,000 annuity due at 9%

N = 5 x 1 = 5 + 1 = 6

R = 9%/1 = 9%

7.5233 x $2,000

$15,046.60 - $2,000

$13,046.60

Manual Calculation2,000.00$ Beginning Yr 1

180.00 9% interest2,180.00

1-20



N = 5 x 1 = 5 + 1 = 6

R = 9%/1 = 9%

7.5233 x $2,000

$15,046.60 - $2,000

$13,046.60


180.00 2,180.00 2,000.00 Beginning Yr 24,180.00

376.20 9 % interest4,556.20

1-21



N = 5 x 1 = 5 + 1 = 6

R = 9%/1 = 9%

7.5233 x $2,000

$15,046.60 - $2,000

$13,046.60


180.00 2,180.00 2,000.00 Beginning Yr 24,180.00

376.20 4,556.20 2,000.00 Beginning Yr 36,556.20

590.06 9 % interest7,146.26

1-22



N = 5 x 1 = 5 + 1 = 6

R = 9%/1 = 9%

7.5233 x $2,000

$15,046.60 - $2,000

$13,046.60


180.00 2,180.00 2,000.00 Beginning Yr 24,180.00

376.20 4,556.20 2,000.00 Beginning Yr 36,556.20

590.06 7,146.26 2,000.00 Beginning Yr 49,146.26

823.16 9 % interest9,969.42

1-23



N = 5 x 1 = 5 + 1 = 6

R = 9%/1 = 9%

7.5233 x $2,000

$15,046.60 - $2,000

$13,046.60


180.00 2,180.00 2,000.00 Beginning Yr 24,180.00

376.20 4,556.20 2,000.00 Beginning Yr 36,556.20

590.06 7,146.26 2,000.00 Beginning Yr 49,146.26

823.16 9,969.42 2,000.00 Beginning Yr 5

11,969.42$ 1,077.25 9 % interest

13,046.67$ End of Yr. 5

1-24

Investing for the Future

• Deb has just started a new job where her employer will match contributions she makes to 401(k) retirement plan. Deb decides to invest $100 per month, with deposits made on a quarterly basis. The company investment plan compounds interest quarterly, and is currently showing a 6% yield. How much will accumulate in Deb’s account after 12 years if this is an ordinary annuity?

• N= 4 * 12 = 48• R = .06/4 = .015• f = 69.5649• FV = 600 * 69.5649 = 41,738.94

1-25

Investing for the Future

• Deb has just started a new job where her employer will match contributions she makes to 401(k) retirement plan. Deb decides to invest $100 per month, with deposits made on a quarterly basis. The company investment plan compounds interest quarterly, and is currently showing a 6% yield. How much will accumulate in Deb’s account after 12 years if this is an annuity due?

• N= 4 * 12 = 48 + 1• R = .06/4 = .015• f = 71.6084• FV = 600 * 71.6084 = 42,965.04 – 600 = 42,365.04

1-26

Assignment

• Read chapter 13 pages 299—305• Work Drill problems 13-1 to 13-4,

page 311

1-27

Annuity Review 1

• Doug Anders gets paid salary plus commission. His commission is paid at the end of each quarter, so he has decided to invest $500 in a mutual fund each quarter. The fund pays 8% interest compounded quarterly. How much will Doug have accumulated at the end of 8 years?

• N = 32 R= 2%• Annuity factor = 44.2269• Ordinary annuity = 44.2269 x 500 = 22113.45

1-28

Annuity review 2

• Nancy Smith receives a $1,250 alimony payment at the beginning of each month. She wants to save for a down payment on a home. If she can save one-half of the alimony each month, and invest it in a fund which pays 6% interest compounded monthly, how much will she have saved after 4 years?

• N = 48 + 1 R = .5%• annuity factor = 55.3684• Annuity due = 55.3684 x 625 = 34605.25 – 625• Annuity due = 33980.25

1-29

• Calculate the present value of an ordinary annuity by table lookup and manually check the calculation

• Compare the calculation of the present value of one lump sum versus the present value of an ordinary annuity


Present Value of an Ordinary Annuity (Find Present Value)

LU13.2LU13.2

1-30

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$3.50

1 2 3

End of period

$2.58

Figure 13.2 - Present value of an annuity of $1 at 8%

1-31

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$3.50

1 2 3

End of period

$1.78

$2.58


1-32

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$3.50

1 2 3

End of period

$.93

$1.78

$2.58


1-33

Calculating Present Value of an Ordinary Annuity by Table Lookup

Step 1. Calculate the number of periods and rate per period

Step 2. Look up the periods and rate in an ordinary annuity table. The intersection gives the table factor for the present value of $1

Step 3. Multiply the withdrawal for each period by the table factor. This gives the present value of an ordinary annuity

1-34

Period 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091

2 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355

3 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869

4 3.8077 3.7171 3.6299 3.5459 3.4651 3.3872 3.3121 3.2397 3.1699

5 4.7134 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908

6 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553

7 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684

8 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349

9 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590

10 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446

11 9.7868 9.2526 8.7605 8.3064 7.8869 7.4987 7.1390 6.8052 6.4951

12 10.5753 9.9540 9.3851 8.8632 8.3838 7.9427 7.5361 7.1607 6.8137

13 11.3483 10.6350 9.9856 9.3936 8.8527 8.3576 7.9038 7.4869 7.1034

14 12.1062 11.2961 10.5631 9.8986 9.2950 8.7455 8.2442 7.7862 7.3667

15 12.8492 11.9379 11.1184 10.3796 9.7122 9.1079 8.5595 8.0607 7.6061

Present value of an annuity of $1 (Partial)

Table 13.2 - Present Value of an Annuity of $1

1-35

Present Value of an AnnuityDuncan Harris wants to receive a $5,000 annuity payment each 6 months for 5 years. Interest on the annuity is 8% semiannually. Duncan will make withdrawals at the end of each year. How much must Duncan invest today to receive a stream of payments for 5 years.

N = 5 x 2 = 10

R = 8%/2 = 4%

8.1109 x $5,000

$40,554.50

Manual Calculation40,554.50$ 1,622.18

42,176.68 (5,000.00) 37,176.68 1,487.07

38,663.75 (5,000.00) 33,663.75 1,346.55

35,010.30 (5,000.00) 30,010.30 1,200.41

31,210.71 (5,000.00) 26,210.71 1,048.43

27,259.14 (5,000.00)

Interest ==>

Payment ==>

Payment ==>

Payment ==>

Payment ==>

Payment ==>

1-36

Present Value of an AnnuityDuncan Harris wants to receive a $5,000 annuity payment each 6 months for 5 years. Interest on the annuity is 8% semiannually. Duncan will make withdrawals at the end of each year. How much must Duncan invest today to receive a stream of payments for 5 years.

N = 5 x 2 = 10

R = 8%/2 = 4%

8.1109 x $5,000

$40,554.50

Manual Calculation40,554.50$ 22,259.14 1,622.18 890.37

42,176.68 23,149.50 (5,000.00) (5,000.00) 37,176.68 18,149.50 1,487.07 725.98

38,663.75 18,875.48 (5,000.00) (5,000.00) 33,663.75 13,875.48 1,346.55 555.02

35,010.30 14,430.50 (5,000.00) (5,000.00) 30,010.30 9,430.50 1,200.41 377.22

31,210.71 9,807.72 (5,000.00) (5,000.00) 26,210.71 4,807.72 1,048.43 192.31

27,259.14 5,000.03 (5,000.00) (5,000.00)

0.03

Interest ==>

Payment ==>

End of Year 5 ==>

Payment ==>

Payment ==>

Payment ==>

Payment ==>

1-37

If I win the Lottery…

1-38

If I win the Lottery…

• Amanda must decide how she wishes to take her lottery winnings, which amount to $1.5 million. She may take a lump sum payment now or her second option is to take an annual payment of $50,400 each year for 20 years, less taxes. Ryan is trying to convince her to take it all now, in one lump-sum. How much would Amanda receive before tax if she follows Ryan’s advice, at a 4% investment rate, annual compounding for 20 years?

• N = 20• R = 4%• PV = 50400 * 13.5903 = $684,951.12

1-39

Lump Sums versus AnnuitiesKaren Jones made deposits of $1,000 to Fleet Bank, which pays 6% interest compounded annually. After 4 years, Karen makes no more deposits. What will be the balance in the account 10 years after the last deposit

N = 4 x 1 = 4

R = 6%/1 = 6%

4.3746 x $1,000

$4,374.60

N = 10 x 1 = 10

R = 6%/1 = 6%

1.7908 x $4,374.60

$7,834.03

Future value of

an annuity

Future value of a lump

sum

1-40

Assignment

• Read chapter 13 pages 306--308• Work Drill problems 13-5 to 13-7,

page 311

1-41

Annuity Review 3

• Susan has started a Roth IRA. She plans to deposit $400 per quarter at the end of each quarter into the fund which pays 6% interest. Tax laws permit Susan to withdraw up to $10,000 for a new home deposit after a minimum of 5 years. How much will Susan have accumulated after the 5 year period?

• N = 5 x 4 = 20 R = 6% / 4 = 1.5%• fv = 400 x 23.1236• fv = 9249.44

1-42

Annuity review 4

• John has just inherited a large sum of money. He wants to set up an educational trust for his daughter. She will start college next year, and he plans on having $5,000 available to pay her educational costs each semester for 5 years. If John can invest the money at 8% compounded semiannually, how much should he put in the trust for his daughter now?

• N = 10 R = 4%• pv = 5000 x 8.1109• pv = 40554.50

1-43

Annuity Review 5

• The YMCA has just received an endowment of $75,000 per year for 4 years, receiving it at the beginning of each year. They plan to invest this in a building fund which will yield 5% annually. At the beginning of the 5th year, they will invest the total amount in a fund for 6 more years, earning 8% quarterly. How much be available after 10 years?

• Annuity due: N = 4 + 1 R = 5%• 75000 x 5.5256 = 414,420 – 75000 = 339,420• Lump sum: N = 24 R = 2%• 339,420 x 1.6084 = 545,923.13

1-44

• Calculate the payment made at the end of each period by table lookup

• Check table lookup by using ordinary annuity table


Sinking Funds (Find Periodic PaymentsLU13.3LU13.3

1-45

Sinking Funds (Find Periodic Payments)

Sinking Fund = Future x Sinking Fund Payment Value Table Factor

Bonds

The amount of a periodic payment which must be made at regular intervals in order to accumulate a desired future amount with compound interest.

1-46

Period 2% 3% 4% 5% 6% 8% 10%

1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2 0.4951 0.4926 0.4902 0.4878 0.4854 0.4808 0.4762

3 0.3268 0.3235 0.3203 0.3172 0.3141 0.3080 0.3021

4 0.2426 0.2390 0.2355 0.2320 0.2286 0.2219 0.2155

5 0.1922 0.1884 0.1846 0.1810 0.1774 0.1705 0.1638

6 0.1585 0.1546 0.1508 0.1470 0.1434 0.1363 0.1296

7 0.1345 0.1305 0.1266 0.1228 0.1191 0.1121 0.1054

8 0.1165 0.1125 0.1085 0.1047 0.1010 0.0940 0.0874

9 0.1025 0.0984 0.0945 0.0907 0.0870 0.0801 0.0736

10 0.0913 0.0872 0.0833 0.0795 0.0759 0.0690 0.0627

11 0.0822 0.0781 0.0741 0.0704 0.0668 0.0601 0.0540

12 0.0746 0.0705 0.0666 0.0628 0.0593 0.0527 0.0468

13 0.0681 0.0640 0.0601 0.0565 0.0530 0.0465 0.0408

14 0.0626 0.0585 0.0547 0.0510 0.0476 0.0413 0.0357

15 0.0578 0.0538 0.0499 0.0463 0.0430 0.0368 0.0315

Sinking fund table based on $1 (Partial)

Table 13.3 - Sinking Fund Table Based on $1

1-47

Sinking Fund

To retire a bond issue, Randolph Company needs $150,000 in 10 years. The interest rate is 8% compounded annually. What payment must Randolph Co. make at the end of each year to meet its obligation?

N = 10 x 1 = 10

R = 8%/1 = 8%

0.0690 x $150,000

$10,350* Off due to rounding

N = 10, R= 8%

Future Value of an annuity

table

$10,350 x 14.4866

149,936.30*

Check:

1-48

If I win the Lottery!!!

• John wants his winnings spread out so he doesn’t spend it all at once. If he wins the megabucks jackpot of $5 million, how much will he receive semi-annually for 20 years, assuming a 4% compound rate?

• SF = 5,000,000 * f• SF = 5,000,000 * .0166• SF = $83,000 before taxes

1-49

Assignment

• Read chapter 13 pages 308--309• Work Drill problems 13-8 to 13-10,

page 311

1-50

Decisions, decisions, decisions …

• How do I know what compound interest factoring table to use for a problem?

1-51

Compound Interest

• to find Future Value (chapter 12)• Given: a single investment amount• What will $1 today grow to in the

future• Present value is known• key words:

• invest today• future amount• at the end of X years

1-52

Compound Interest

• to find Present Value (chapter 12—col 2)

• Given: a single future amount

• What you need today to grow to $1 in the future

• Future value is known• key words:

• how much today• present value

1-53

Annuity—Future Value• to find Future Value of several payments made

(chapter 13)• Amount of Annuity (col. 3)• multiple amounts grow to one future amount• What will several $1 deposits grow to in the future• Given: payment amounts• key words:

• annuity• end of X years • Payments, Deposits • Ordinary Annuity• Annuity Due

1-54

Annuity—Present Value

• to find amount needed today to receive several equal payments (chapter 13)

• Present Value of Annuity (col. 4)• Given: amount of multiple payments• What will you need today (single

amount) if you wish to make multiple withdrawal over future periods

• key words:• how much today• original amount• present value

1-55

Annuity—Sinking Fund

• to find amount of each equal payment (chapter 13)

• Sinking Fund (col. 5)• How much is the payment?• Given: Future amount needed• key words:

• needed payments• deposits needed• sinking fund payment

1-56

Review Problem 1

• John Tobn made deposit of $650 at the end of each year for 5 years. The rate if 8% compounded annually. What is the value of John’s annuity at the end of 5 years?

• What type of problem?• Ordinary annuity• N = 5• R = .08• 650 x 5.8666 = 3813.29

1-57

Review Problem 2

• Jim promised to pay his son $200 semiannually for 6 years. If Jim can invest his money at 8% in an ordinary annuity, how much must Jim invest today to be able to pay his son.

• What type of problem?• Annuity—present value• N = 12• R = .04• 200 * 9.3851 = $1877.02

1-58

Review Problem 3• Joyce has decided to invest $500 quarterly for 5 years

in an ordinary annuity at 12%. As her financial advisor, could you calculate for Joyce the total cash value of the annuity at the end of year 5?

• What type of problem?• Ordinary annuity• N = 20• R = 3%• 500 * 26.8704 = 13,435.20

1-59

Review Problem 4

• Jack invests $850 semiannually at 10% for 9 years at the beginning of each year. What will be the cash value of this annuity due at the end of the ninth year?

• What type of problem?• Annuity due• N = 19• R = 5%• 850 * 30.5389 = $25,958.07 – 850 = 25,108.07

1-60

Review Problem 5• Moller Associates must repay $500,000 in bonds in 10

years. The company wants to set up a sinking fund to accumulate the needed amount. Assuming a rate of 10% compounded semiannually, what amount must be paid into the fund each period?

• What type of problem?• Sinking fund• N = 20 R = 5%• 500,000 * .0302 = 15,100• Check it:• 15100 * 33,0659 = 499,295.10

1-61

Review Problem 6• Jane’s daughter, Amanda, is now two year’s old. Jane is planning

a college fund to pay tuition costs of $5000 per semester. Amanda will start college in 16 years, and Jane will pay for four years of school. Jane is planning on setting up a mutual fund which yields 7% compounded semiannually, using money she inherited. How much must Jane invest into the mutual fund to meet the needed tuition payments?

• What is this? • Present value of an annuity and sinking fund• PV annuity: 5000, 4 years, semiannual, 7%• PV annuity = 5000 x 6.8739 = 34,369.50 in 16 years• PV lump sum for 16 years at 7% semiannual• PV = 34,369 * .3326 = 1,1431.13

1-62

Review Problem 7

• Wes wants to retire as a millionaire. How much must Wes save per year for 30 years, if he can invest in a mutual fund which pays 9% compounded annually?

• What is this?• Sinking fund• SF = 1,000,000 x .0073• SF deposit = 7300 per year

Documents

1-1. 1-2 Chapter 13 Annuities and Sinking Funds McGraw-Hill/Irwin Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved