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

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Chapter 13 Annuities and

Annuities and

Sinking Funds

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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

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Learning Unit Objectives

Annuities: Ordinary Annuity and

Annuity Due (Find Future Value)

LU13.1

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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

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Learning Unit Objectives

Present Value of an Ordinary Annuity

(Find Present Value)

LU13.2

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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

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Learning Unit Objectives

Sinking Funds (Find Periodic Payments

LU13.3

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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

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$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%

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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

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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

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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 end of the desired period is reached.

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

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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 Calculation 3,000.00 $ 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

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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 annuity ordinary annuity each period table factor

Calculating Future Value of an Ordinary

Annuity by Table Lookup

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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

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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%

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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.

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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 Calculation 3,000.00 $ Beginning Yr 1 240.00 3,240.00 3,000.00 Beginning Yr 2 6,240.00 499.20 6,739.20 3,000.00 Beginning Yr 3 9,739.20 779.14 10,518.34 End of Yr. 3

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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.

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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

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$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%

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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 of ordinary annuity pymt. Pymt. ordinary annuity table

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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

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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 Calculation 20,616.80 $ 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 ==>

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Lump Sums versus Annuities

John 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

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Lump Sums versus Annuities

Mel 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 2

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Sinking Funds (Find Periodic Payments)

Bonds

Sinking Fund = Future x Sinking Fund Payment Value Table Factor

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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