The main contents of the chapter consist of the following: Annuities: ordinary annuity and annuity due (find future value), present value of an ordinary annuity (find present value), sinking funds (find periodic payments).
Chapter Thirteen Annuities and Sinking Funds McGrawHill/Irwin Copyright © 2014 by The McGrawHill Companies, Inc. All rights reserved 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 LU 13-2: 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 LU 13-3: 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 132 Compounding Interest (Future Value) Annuity – Term of the annuity – A series of payments 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 – Tthe amount of money needed to invest today in order to receive a stream of payments for a given number of years in the future 133 Future value of an annuity of $1 at 8% (Figure 13.1) $3.2464 $3.50 $3.00 $2.0800 $2.50 $2.00 $1.50 $1.00 $1.00 $0.50 $0.00 End of period 134 Classification of Annuities Contingent annuities – Annuities certain – have no fixed number of payments but depend on an uncertain event have a specific stated number of payments Life Insurance payments Mortgage payments 135 Classification of Annuities Ordinary annuity – Annuity due – regular deposits/payments made at the end of the period 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. 1 136 Calculating Future Value of an Ordinary Annuity Manually Step For period 1, no interest calculation is necessary, since money is invested at the end of the period Step For period 2, calculate interest on the balance and add the interest to the previous balance Step Add the additional investment at the end of period to the new balance Step Repeat Steps and until the end of the desired period is reached 137 Calculating Future Value of an Ordinary Annuity Manually Find the value of an investment after years for a $3,000 ordinary annuity at 8% Manual Calculation $ 3,000.00 End of Yr 240.00 plus interest 3,240.00 3,000.00 Yr Investment 6,240.00 End of Yr 499.20 plus interest 6,739.20 3,000.00 Yr Investment $ 9,739.20 End of Yr 138 Calculating Future Value of an Ordinary Annuity by Table Lookup Step Calculate the number of periods and rate per period Step Look up the periods and rate in an ordinary annuity table The intersection gives the table factor for the future value of $1 Step Multiply the payment each period by the table factor This gives the future value of the annuity Future value of annuity ordinary annuity = Annuity payment x each period Ordinary table factor 139 Ordinary annuity table: Compound sum of an annuity of $1 (Table 13.1) Ordinary annuity table: Compound sum of an annuity of $1 (partial) Period 2% 3% 4% 5% 6% 7% 8% 9% 10% 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.0200 2.0300 2.0400 2.0500 2.0600 2.0700 2.0800 2.0900 2.1000 3.0604 3.0909 3.1216 3.1525 3.1836 3.2149 3.2464 1.0000 3.3100 4.1216 4.1836 4.2465 4.3101 4.3746 4.4399 4.5061 4.5731 4.6410 5.2040 5.3091 5.4163 5.5256 5.6371 5.7507 5.8666 5.9847 6.1051 6.3081 6.4684 6.6330 6.8019 6.9753 7.1533 7.3359 7.5233 7.7156 7.4343 7.6625 7.8983 8.1420 8.3938 8.6540 8.9228 9.2004 9.4872 8.5829 8.8923 9.2142 9.5491 9.8975 10.2598 10.6366 11.0285 11.4359 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 1310 Future Value of an Ordinary Annuity Find the value of an investment after years for a $3,000 ordinary annuity at 8% Periods (N) = x = Rate (R) = 8%/1 = 8% 3.2464 (table factor) x $3,000 = $9,739.20 1311 Calculating Future Value of an Annuity Due Manually Step Calculate the interest on the balance for the period and add it to the previous balance Step Add additional investment at the beginning of the period to the new balance Step Repeat Steps and until the end of the desired period is reached 1312 Calculating Future Value of an Annuity Due Manually Find the value of an investment after years for a $3,000 annuity due at 8% Manual Calculation $ 3,000.00 Beginning Yr 240.00 Yr Interest 3,240.00 3,000.00 Beginning Yr 6,240.00 499.20 Yr Interest 6,739.20 3,000.00 Beginning Yr 9,739.20 779.14 Yr Interest 10,518.34 End of Yr 1313 Calculating Future Value of an Annuity Due by Table Lookup Step Calculate the number of periods and rate per period Add one extra period Step Look up in an ordinary annuity table the periods and rate The intersection gives the table factor for the future value of $1 Step Multiply the payment each period by the table factor Step Subtract payment from Step 1314 Future Value of an Annuity Due Find the value of an investment after years for a $3,000 annuity due at 8% Periods (N) = x = + = Rate (R) = 8%/1 = 8% 4.5061 (table factor) x $3,000 = $13,518.30 $13,518.30 $3,000 = $10,518.30 1315 Present value of an annuity of $1 at 8% (Figure 13.2) $3.50 $2.5771 $3.00 $2.50 $1.7833 $2.00 $1.50 $.9259 $1.00 $0.50 $0.00 Number of periods 1316 Calculating Present Value of an Ordinary Annuity by Table Lookup Step Calculate the number of periods and rate per period Step Look up the periods and rate in the present value of an annuity table The intersection gives the table factor for the present value of $1 Step Multiply the withdrawal for each period by the table factor This gives the present value of an ordinary annuity Present value of ordinary annuity payment = Annuity x payment Present value of ordinary annuity table 1317 Present Value of an Annuity of $1 (Table 13.2) Present value of an annuity of $1 (partial) Period 2% 3% 4% 5% 6% 7% 8% 9% 10% 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869 3.8077 3.7171 3.6299 3.5459 3.4651 3.3872 3.3121 3.2397 3.1699 4.7134 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349 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 1318 Present Value of an Annuity John Fitch wants to receive a $8,000 annuity in 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 years N=3x1=3 periods R = 8%/1 = 8% 2.5771 (table factor) x $8,000 = $20,616.80 Manual Calculation $ 20,616.80 1,649.34 Interest ==> 22,266.14 (8,000.00) Payment ==> 14,266.14 1,141.29 Interest ==> 15,407.43 (8,000.00) Payment ==> 7,407.43 592.59 Interest ==> 8,000.02 (8,000.00) Payment ==> 0.02 End of Year 3 ==> 1319 Lump Sums versus Annuities John Sands made deposits of $200 to Floor Bank, which pays 8% interest compounded annually After years, John makes no more deposits What will be the balance in the account years after the last deposit? Step Future value of an annuity N = x = 10 periods R = 8%/2 = 4% 12.0061 (table factor) x $200 = $2,401.22 Step Future value of a lump sum N = x = 12 periods R = 8%/2 = 4% 1.6010 (table factor) x $2,401.22 = $3,844.35 1320 Lump Sums versus Annuities Mel Rich decided to retire in 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 Step Present value of an annuity N = 25 x = 25 periods R = 5%/1 = 5% Step Present value of a lump sum N = x = periods R = 5%/1 = 5% 6768 x $563,756 = $381,550.06 14.0939 x $40,000 = $563,756 1321 Sinking Funds (Find Periodic Payments) Sinking fund – financial arrangement that sets aside regular periodic payments of a particular amount of money Sinking fund = Future x Sinking fund payment value table factor 1322 SINKING FUND TABLE BASED ON $1 (Table 12.3) Period 2% 3% 4% 5% 6% 8% 10% 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.4951 0.4926 0.4902 0.4878 0.4854 0.4808 0.4762 0.3268 0.3235 0.3203 0.3172 0.3141 0.3080 0.3021 0.2426 0.2390 0.2355 0.2320 0.2286 0.2219 0.2155 0.1922 0.1884 0.1846 0.1810 0.1774 0.1705 0.1638 0.1585 0.1546 0.1508 0.1470 0.1434 0.1363 0.1296 0.1345 0.1305 0.1266 0.1228 0.1191 0.1121 0.1054 0.1165 0.1125 0.1085 0.1047 0.1010 0.0940 0.0874 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 1323 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 = 18 periods R = 10%/1 = 10% 0.0219 x $60,000 = $1,314 Check Future Value of an annuity table N = 18, R= 10% $1,314 x 45.5992 = $59,917.35* * Off due to rounding 1324 ... LU1 3-1 : 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. .. $563,756 1321 Sinking Funds (Find Periodic Payments) Sinking fund – financial arrangement that sets aside regular periodic payments of a particular amount of money Sinking fund = Future x Sinking fund... of the present value of one lump sum versus the present value of an ordinary annuity LU 1 3-3 : Sinking Funds (Find Periodic Payments) Calculate the payment made at the end of each period by table