Chapter 13

Annuities and Annuities and Sinking FundsSinking Funds

Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

13-2

1. Differentiate between contingent annuities and annuities certain

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

Annuities and Sinking Funds#13#13Learning Unit ObjectivesAnnuities: Ordinary Annuity and Annuity Due (Find Future Value)

LU13.1LU13.1

13-3

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

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

Annuities and Sinking Funds#13#13Learning Unit ObjectivesPresent Value of an Ordinary Annuity (Find Present Value)

LU13.2LU13.2

13-4

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

2. Check table lookup by using ordinary annuity table

Annuities and Sinking Funds#13#13Learning Unit ObjectivesSinking Funds (Find Periodic PaymentsLU13.3LU13.3

13-5

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

13-6

$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

$3.2464

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

13-7

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

13-8

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

13-9

Step 1. For period 1, no interest calculation is necessary, since money is invested at the end of period

Step 3. Add the additional investment at the end of period 2 to the new balance.

Calculating Future Value of an Ordinary Annuity Manually

Step 4. Repeat steps 2 and 3 until the endof the desired period is reached.

Step 2. For period 2, calculate interest on the balance and add the interest to the previous balance.

13-10

Calculating Future Value of an Ordinary Annuity Manually

Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%

Manual Calculation3,000.00$ End of Yr 1

240.00 3,240.00 3,000.00 6,240.00 End of Yr 2

499.20 6,739.20 3,000.00 9,739.20 End of Yr 3

13-11

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.Future value of = Annuity pymt. x Ordinary annuityordinary annuity each period table factor

Calculating Future Value of an Ordinary Annuity by Table Lookup

13-12

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

13-13

N = 3 x 1 = 3

R = 8%/1 = 8%

3.2464 x $3,000

$9,739.20

Future Value of an Ordinary Annuity

Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%

13-14

Calculating Future Value of an Annuity Due Manually

Step 1. Calculate the interest on the balance for the period and add it to the previous balance

Step 2. Add additional investment at the beginning of the period to the new balance.

Step 3. Repeat steps 1 and 2 until the end of the desired period is reached.

13-15

Calculating Future Value of an Annuity Due Manually

Find the value of an investment after 3 years for a $3,000 annuity due at 8%

Manual Calculation3,000.00$ Beginning Yr 1

240.00 3,240.00 3,000.00 Beginning Yr 26,240.00

499.20 6,739.20 3,000.00 Beginning Yr 39,739.20

779.14 10,518.34 End of Yr. 3

13-16

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.

13-17

Future Value of an Annuity Due

Find the value of an investment after 3 years for a $3,000 annuity due at 8% N = 3 x 1 = 3 + 1 = 4

R = 8%/1 = 8%

4.5061 x $3,000

$13,518.30 - $3,000

$10,518.30

13-18

$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

$2.5771

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

13-19

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 Present value of = Annuity x Present value ofordinary annuity pymt. Pymt. ordinary annuity table

13-20

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

13-21

Present Value of an Annuity

John Fitch wants to receive a $8,000 annuity in 3 years. Interest on the annuity is 8% semiannually. John will make withdrawals at the end of each year. How much must John invest today to receive a stream of payments for 3 years.

N = 3 x 1 = 3

R = 8%/1 = 8%

2.5771 x $8,000

$20,616.80

Manual Calculation20,616.80$ 1,649.34

22,266.14 (8,000.00) 14,266.14 1,141.29

15,407.43 (8,000.00) 7,407.43

592.59 8,000.02

(8,000.00) 0.02

Interest ==>

Payment ==>

End of Year 3 ==>

Interest ==>

Interest ==>

Payment ==>

Payment ==>

13-22

Lump Sums versus AnnuitiesJohn Sands made deposits of $200 to Floor Bank, which pays 8% interest compounded annually. After 5 years, John makes no more deposits. What will be the balance in the account 6 years after the last deposit?

N = 5 x 2 = 10

R = 8%/2 = 4%

12.0061 x $200

$2,401.22

N = 6 x 2 = 12

R = 8%/2 = 4%

1.6010 x $2,401.22

$3,844.35

Future value of

an annuity

Future value of a lump

sum

Step 1

Step 2

13-23

Lump Sums versus AnnuitiesMel Rich decided to retire in 8 years to New Mexico. What amount must Mel invest today so he will be able to withdraw $40,000 at the end of each year 25 years after he retires? Assume Mel can invest money at 5% interest compounded annually.

N = 25 x 1 = 25

R = 5%/1 = 5%

14.0939 x $40,000

$563,756

N = 8 x 1 = 8

R = 5%/1 = 5%

.6768 x $563,756

$381,550.06

Present value of

an annuity

Present value of a lump sum

Step 1

Step 2

13-24

Sinking Funds (Find Periodic Payments)

Bonds

Sinking Fund = Future x Sinking Fund Payment Value Table Factor

Bonds

13-25

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

16 0.0537 0.0496 0.0458 0.0423 0.0390 0.0330 0.0278

17 0.0500 0.0460 0.0422 0.0387 0.0354 0.0296 0.0247

18 0.0467 0.0427 0.0390 0.0355 0.0324 0.0267 0.0219

Table 13.3 - Sinking Fund Table Based on $1

13-26

Sinking Fund

To retire a bond issue, Moore Company needs $60,000 in 18 years from today. The interest rate is 10% compounded annually. What payment must Moore make at the end of each year? Use Table 13.3.

N = 18 x 1 = 18

R = 10%/1 = 10%

0.0219 x $60,000

$1,314

Check

$1,314 x 45.5992

59,917.35*

* Off due to rounding

N = 18, R= 10%

Future Value of an annuity table

13-27

Problem 13-13:

18 periods + 1 = 19, 5%

30.5389X $2,000$61,077.80-$ 2,000.00-$59,077.80

Solution:

13-28

Problem 13-17:

20 periods, 12% (Table 13.1)

$12,500 x 72.0524 = $900,655

Solution:

13-29

Problem 13-18:

10 periods, 11% (Table 13.2)

$15,000 x 5.8892 = $88,338

Solution:

13-30

Problem 13-23:

16 periods, = 2%8% 4

$900 x 13.577 = $12,219.93

OR

$900 x 18.6392 = $16,775.28 x .7284 (Table 12.3)

$12,219.11 2% 16 periods

Solution:

13-31

Problem 13-25:

20 periods, 2% (Table 13.3)

.0412 x $88,000 = $3,625.60 quarterly payment

Solution:

13-32

Problem 13-26:

Morton: 5 periods, 8%

3.9927 x $35,000 = $139,744.50 + $40,000 = $179,744.50

Flynn: 5 periods, 8%

3.9927 x $38,000 = $151,722.60 + $25,000 = $176,722.60

Morton offered a better.

Solution:

13-33

Problem 13-27:

PV annuity table: 15 periods, 8% 8.5595 x $28,000 = $239,666

PV table: 10 years, 8% .4632 x $239,666 = $111,013.29

Solution: