Chapter 13 Annuities and Sinking Funds 13-1 McGraw-Hill/Irwin Copyright 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Compounding Interest (Future Value) Annuity - A series of payments--can be payments going out or coming in--just a series of payments (in the financial community, this is called a stream of payments). Term of the annuity - the time from the beginning of the first payment period to the end of the last payment period. 13-2
Future Value of an Annuity The future dollar amount of a series of payments plus interest (You want to know how much you will end up with.) Example: You know you can afford to save $400 each period; you know the interest rate; you want to know how much you will have at the end of a specific period of time. 13-3
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. (You want to know how much to put in the bank so that you can start taking out fixed sums of money in the future. ) Example: How much do your parents put in the bank, at a specific interest rate, on your first day of college so that you can take out $2,000 each semester over the course of four years in college so that it lasts until the end of the fourth year? 13-4
Sinking fund: A kind of annuity. You know the amount you need at a certain date in the future. You are looking for the amount of the payment in a stream of payments in order to end up with that amount. Example: You will need $25,000 five years in the future to replace your car. You want to know how much to put in the bank each month in order to end up with $25,000 in five years. You know the interest rate and number of periods you have to save the money. The $25,000 that you will ultimately have is made up of your stream of payments into the sinking fund (in this case, a savings account) and interest payments on the money in that account over the five years. 13-5
Figure 13.1 Future value of an annuity of $1 at 8% $3.50 $3.25 $3.00 $2.50 $2.00 $2.08 $1.50 $1.00 $1.00 $0.50 $0.00 1 2 3 End of period 13-6
Classification of Annuities This is Important! 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 March 30 Quarterly Jan. 1 June 30 Semiannually Jan. 1 Dec. 31 Annually Jan. 1 13-7
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 Annually: 3 x 1 = 3 Semiannually: 3 x 2 = 6 Quarterly: 3 x 4 = 12 Rate Annually: 6% / 1 = 6% Semiannually: 6% / 2 = 3% Quarterly: 6% / 4 = 1.5% 13-8
Calculating Future Value of an Ordinary Annuity Manually Step 1. For period 1, no interest calculation is necessary, since money is invested at the end of period Step 2. For period 2, calculate interest on the balance and add the interest to the previous balance. Step 3. Add the additional investment at the end of period 2 to the new balance. Step 4. Repeat steps 2 and 3 until the end of the desired period is reached. 13-9
13-10 Calculating Future Value of an Ordinary Annuity by Hand (deposits at ends of years) Find the value of an investment after 5 years for a $2,000 ordinary annuity at 9% Manual Calculation $ 2,000.00 End of Yr 1 180.00 int, yr 2 2,180.00 new bal. 2,000.00 End of Yr 2 4,180.00 new bal. 376.20 int, yr 3 4,556.20 new bal. 2,000.00 End of Yr 3 6,556.20 new bal. 590.06 int, yr 4 7,146.26 new bal. 2,000.00 End of Yr 4 9,146.26 new bal. 823.16 int, yr 5 9,969.42 new bal. 2,000.00 End of Yr 5 $ 11,969.42 new bal.
Calculating Future Value of an Ordinary Annuity by Table Lookup Step 1. Calculate the number of periods and rate per period. Step 2. Look up the number periods on the appropriate interest rate page in the Math Handbook. When you find the row for the correct number of periods, follow across that row to the amount of annuity column. The intersection gives the table factor for the future value. Step 3. Multiply the payment for a period by the table factor. This gives the future value of the annuity. Future value of = Annuity pmt. x Ordinary annuity ordinary annuity for a period table factor 13-11
Table 13.1 Ordinary annuity table: Compound sum of an annuity of $1 Ordinary annuity table: Compound sum of an annuity of $1 (Partial) 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 13-12 15 17.2934 18.5989 20.0236 21.5785 23.2759 25.1290 27.1521 29.3609 31.7725
9% Looking Up the Future Value of an Annuity--9%, 5 periods---as shown in the Math Handbook Period 13-13 Compound Interest (future value) Present Value Amount of Annuity (future value) Present Value of Annuity Sinking Fund 1 1.0900 0.9174 1.0000 0.9174 1.0000 2 1.1881.8417 2.0900 1.7591 0.4785 3 1.2950 0.7722 3.2781 2.5313 0.3051 4 1.4116 0.7084 4.5731 3.2397 0.2187 5 1.5386 0.6499 5.9847 3.8897 0.1671 6 1.6771 0.5963 7.5233 4.4859 0.1329 7 1.8280 0.5470 9.2004 5.0330 0.1087 8 1.9926 0.5019 11.0285 5.5348 0.0907 9 2.1719 0.4604 13.0210 5.9952 0.0768 10 2.3674 0.4224 15.1929 6.4177 0.0658
Future Value of an Ordinary Annuity Find the value of an investment after 5 years for a $2,000 ordinary annuity at 9% N = 5 x 1 = 5 R = 9%/1 = 9% 5.9847 x $2,000 $11,969.40 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. See example on next slide. 13-15
13-16 Calculating Future Value of an Annuity Due manually (deposits at beginning of period) Find the value of an investment after 5 years for a $2,000 annuity due at 9% YEAR 1 2,000.00 deposit, beginning year 1 180.00 interest earned during year 1 2,180.00 new balance at end of year 1 YEAR 2 2,000.00 deposit, beginning year 2 4,180.00 balance after year 2 deposit 376.20 interest earned during year 2 4,556.20 new balance at end of year 2 YEAR 3 2,000.00 deposit, beginning year 3 6,556.20 balance after year 3 deposit 590.06 interest earned during year 3 7,146.26 new balance at end of year 3 YEAR 4 2,000.00 deposit, beginning year 4 9,146.26 balance after year 4 deposit 823.16 interest earned during year 4 9,969.42 new balance at end of year 4 YEAR 5 2,000.00 deposit, beginning year 5 11,969.42 balance after year 5 deposit 1,077.25 interest earned during year 5 13,046.67 new balance at end of year 5
Calculating Future Value of an Annuity Due by Table Lookup Calculate the number of periods and rate per period. Add one extra period. Look up the number of periods on the appropriate interest rate page in the Math Handbook. At the intersection of the correct period row and the Amount of Annuity column, you will find the needed table factor. Multiply the payment each period by the table factor. Subtract 1 payment. 13-17
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 13-18
Figure 13.2 - Present value of an annuity of $1 at 8% $3.50 $3.00 $2.58 $2.50 $2.00 $1.78 $1.50 $1.00 $.93 $0.50 $0.00 1 2 3 End of period 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. Find the percent page in the Math Handbook. Look up the number of periods in the first column. At the intersection of the correct period row and the present value of annuity column, you will find the needed table factor. Step 3. Multiply the withdrawal for a period by the table factor. This gives the present value of an ordinary annuity 13-20
Table 13.2 - Present Value of an Annuity of $1 Present value of an annuity of $1 (Partial) 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 13-21 15 12.8492 11.9379 11.1184 10.3796 9.7122 9.1079 8.5595 8.0607 7.6061
Period 5% Compound Interest (future value) Present Value Amount of Annuity (future value) Present Value of Annuity Sinking Fund 1 1.0500 0.9524 1.0000 0.9524 1.0000 2 1.1025 0.9070 2.0500 1.8594 0.4878 3 1.1576 0.8638 3.1525 2.7232 0.3172 4 1.2155 0.8227 4.3101 3.5459 0.2320 5 1.2763 0.7835 5.5256 4.3295 0.1810 6 1.3401 0.7462 6.8019 5.0757 0.1470 7 1.4071 0.7107 8.1420 5.7864 0.1228 8 1.4775 0.6768 9.5491 6.4632 0.1047 9 1.5513 0.6446 11.0265 7.1078 0.0907 10 1.6289 0.6139 12.5779 7.7217 0.0795 13-22 Present Value of an Annuity at 5% and 10 periods
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. 13-23 N = 5 x 2 = 10 R = 8%/2 = 4% 8.1109 x $5,000 $40,554.50 Interest ==> withdrawal-> Interest ==> withdrawal=> Interest ==> withdrawal=> Interest ==> withdrawal=> Interest ==> withdrawal=> End of Year 5 ==> Manual Calculation $ 40,554.50 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
Sinking Funds (Find Periodic Payments) Bonds Bonds Sinking Fund = Future x Sinking Fund Payment Value Table Factor 13-24
Table 13.3 - Sinking Fund Table Based on $1 Sinking fund table based on $1 (Partial) 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 13-25 15 0.0578 0.0538 0.0499 0.0463 0.0430 0.0368 0.0315
Period 8% Compound Interest (future value) Present Value Amount of Annuity (future value) Present Value of Annuity Sinking Fund 1 1.0800 0.9259 1.0000 0.9259 1.0000 2 1.1664 0.8573 2.0800 1.7833.4808 3 1.2597 0.7938 3.2464 2.5771 0.3080 4 1.3605 0.7350 4.5061 3.3121 0.2219 5 1.4693 0.6806 5.8666 3.9927 0.1705 6 1.5869 0.6302 7.3359 4.6229 0.1363 7 1.7138 0.5835 8.9228 5.2064 0.1121 8 1.8509 0.5403 10.6366 5.7466 0.0940 9 1.9990 0.5002 12.4876 6.2469 0.0801 10 2.1589 0.4632 14.4866 6.7101 0.0690 13-26 Sinking Fund at 8% and 10 periods
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? Check $10,350 x 14.4866 149,936.30* N = 10 x 1 = 10 R = 8%/1 = 8% 0.0690 x $150,000 $10,350 N = 10, R= 8% Use Sinking Fund Column * Off due to rounding 13-27