13-1 Chapter 13 Annuities and Sinking Funds 13-2 McGraw-Hill/Irwin Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. #13 Annuities and Sinking Funds Learning Unit Objectives LU13.1 Annuities: Ordinary Annuity and Annuity Due (Find Future Value) • Differentiate between contingent annuities and annuities certain • Calculate the future value of an ordinary annuity and an annuity due manually and by table lookup 13-3 #13 Annuities and Sinking Funds Learning Unit Objectives LU13.2 Present Value of an Ordinary Annuity (Find Present Value) • 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 13-4 #13 Annuities and Sinking Funds Learning Unit Objectives LU13.3 Sinking Funds (Find Periodic Payments • Calculate the payment made at the end of each period by table lookup • Check table lookup by using ordinary annuity table 13-5 Compounding Interest (Future Value) Annuity - A series of payments Future value of annuity the future dollar amount of a series of payments plus interest 13-6 Term of the annuity - the time from the beginning of the first payment period to the end of the last payment period. 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 Figure 13.1 Future value of an annuity of $1 at 8% $3.25 $3.50 $3.00 $2.50 $2.08 $2.00 $1.50 $1.00 $1.00 $0.50 $0.00 1 2 End of period 13-7 3 Classification of Annuities 13-8 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 Classification of Annuities Annuity due regular deposits/payments made at the beginning of the period Ordinary annuity regular deposits/payments made at the end of the period 13-9 Jan. 31 Monthly Jan. 1 March 30 Quarterly Jan. 1 June 30 Semiannually Jan. 1 Dec. 31 Annually Jan. 1 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? Periods Rate Annually: 6% / 1 = 6% Annually: 3x1=3 Semiannually: 6% / 2 = 3% Semiannually: 3 x 2 = 6 Quarterly: 6% / 4 = 1.5% Quarterly: 3 x 4 = 12 13-10 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 3. Add the additional investment at the end of period 2 to the new balance. Step 2. For period 2, calculate interest on the balance and add the interest to the previous balance. Step 1. For period 1, no interest calculation is necessary, since money is invested at the end of period 13-11 Calculating Future Value of an Ordinary Annuity Manually Find the value of an investment after 5 years for a $2,000 ordinary annuity at 9% 13-12 Manual Calculation $ 2,000.00 End of Yr 1 180.00 2,180.00 2,000.00 End of Yr 2 4,180.00 376.20 4,556.20 2,000.00 End of Yr 3 6,556.20 590.06 7,146.26 2,000.00 End of Yr 4 9,146.26 823.16 9,969.42 2,000.00 End of Yr 5 $ 11,969.42 Calculating Future Value of an Ordinary Annuity by Table Lookup 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 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 1. Calculate the number of periods and rate per period 13-13 Table 13.1 Ordinary annuity table: Compound sum of an annuity of $1 Ordinary annuity table: Compound sum of an annuity of $1 (Partial) 13-14 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 Future Value of an Ordinary Annuity Find the value of an investment after 5 years for a $2,000 ordinary annuity at 9% 13-15 N=5x1=5 R = 9%/1 = 9% 5.9847 x $2,000 $11,969.40 Calculating Future Value of an Annuity Due Manually Step 3. Repeat steps 1 and 2 until the end of the desired period is reached. Step 2. Add additional investment at the beginning of the period to the new balance. Step 1. Calculate the interest on the balance for the period and add it to the previous balance 13-16 Calculating Future Value of an Annuity Due Manually Find the value of an investment after 5 years for a $2,000 annuity due at 9% 13-17 Manual Calculation $ 2,000.00 Beginning Yr 1 180.00 2,180.00 2,000.00 Beginning Yr 2 4,180.00 376.20 4,556.20 2,000.00 Beginning Yr 3 6,556.20 590.06 7,146.26 2,000.00 Beginning Yr 4 9,146.26 823.16 9,969.42 2,000.00 Beginning Yr 5 $ 11,969.42 1,077.25 $ 13,046.67 End of Yr. 5 Calculating Future Value of an Annuity Due by Table Lookup Step 4. Subtract 1 payment from Step 3. Step 3. Multiply the payment each period by the table factor. 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 1. Calculate the number of periods and rate per period. Add one extra period. 13-18 Future Value of an Annuity Due Find the value of an investment after 5 years for a $2,000 annuity due at 9% N=5x1=5+1=6 R = 9%/1 = 9% 7.5233 x $2,000 $15,046.60 - $2,000 $13,046.60 13-19 Figure 13.2 - Present value of an annuity of $1 at 8% $3.50 $3.00 $2.58 $2.50 $1.78 $2.00 $1.50 $.93 $1.00 $0.50 $0.00 1 2 End of period 13-20 3 Calculating Present Value of an Ordinary Annuity by Table Lookup 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 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 1. Calculate the number of periods and rate per period 13-21 Table 13.2 - Present Value of an Annuity of $1 Present value of an annuity of $1 (Partial) 13-22 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 Duncan Harris wants to receive a $5,000 annuity in 5 years. Interest on the annuity is 8% semiannually. Duncan will make withdrawals every six months. How much must Duncan invest today to receive a stream of payments for 5 years. Interest ==> Payment ==> Interest ==> Payment ==> Interest ==> Payment ==> N = 5 x 2 = 10 R = 8%/2 = 4% Payment ==> 8.1109 x $5,000 Interest ==> $40,554.50 13-23 Interest ==> Payment ==> End of Year 5 ==> Manual Calculation $ 40,554.50 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) 22,259.14 890.37 23,149.50 (5,000.00) 18,149.50 725.98 18,875.48 (5,000.00) 13,875.48 555.02 14,430.50 (5,000.00) 9,430.50 377.22 9,807.72 (5,000.00) 4,807.72 192.31 5,000.03 (5,000.00) 0.03 Lump Sums versus Annuities Karen 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 Future value of a lump sum N=4x1=4 R = 6%/1 = 6% 4.3746 x $1,000 $4,374.60 Step 1 13-24 Future value of an annuity N = 10 x 1 = 10 R = 6%/1 = 6% 1.7908 x $4,374.60 $7,834.03 Step 2 Sinking Funds (Find Periodic Payments) Bonds Bonds Sinking Fund = Future Payment Value 13-25 x Sinking Fund Table Factor Table 13.3 - Sinking Fund Table Based on $1 Sinking fund table based on $1 (Partial) 13-26 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 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 13-27 Check $10,350 x 14.4866 149,936.30* N = 10, R= 8% Future Value of an annuity table * Off due to rounding