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Chapter Thirteen
ANNUITIES AND SINKING FUNDS
Copyright © 2014 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.
LU13-1: Annuities: Ordinary Annuity and Annuity Due (Find Future Value)
LEARNING UNIT OBJECTIVES
LU 13-2: Present Value of an Ordinary Annuity (Find Present Value)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.
LU 13-3: Sinking Funds (Find Periodic Payments)
1. Calculate the payment made at the end of each period by table lookup.
2. Check table lookup by using ordinary annuity table.
13-3
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 – 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
Annuity –
A series of payments
13-4
$0.00
$0.50
$1.00
$1.50
$2.00
$2.50
$3.00
$3.50
1 2 3End of period
$1.00
$2.0800
$3.2464
FUTURE VALUE OF AN ANNUITY OF $1
AT 8% (FIGURE 13.1)
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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
Life Insurance payments Mortgage payments
13-6
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
13-7
Step 1. For period 1, no interest calculation is necessary, since money is invested at the end of the 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.
CALCULATING FUTURE VALUE OF AN ORDINARY ANNUITY MANUALLY
Step 4. Repeat Steps 2 and 3 until the end of the desired period is reached.
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CALCULATING FUTURE VALUE OF AN ORDINARY ANNUITY MANUALLY
Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%.
Manual Calculation3,000.00$ End of Yr 1
240.00 plus interest3,240.00 3,000.00 Yr. 2 Investment6,240.00 End of Yr 2
499.20 plus interest6,739.20 3,000.00 Yr. 3 Investment9,739.20$ End of Yr 3
13-9
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 future value of $1.
Step 3. Multiply the payment each period by the table factor. This gives the future value of the annuity.
CALCULATING FUTURE VALUE OF AN ORDINARY ANNUITY BY TABLE
LOOKUP
Future value of = Annuity payment x Ordinary annuity
ordinary annuity each period table factor
13-10
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)
ORDINARY ANNUITY TABLE: COMPOUND SUM OF AN ANNUITY OF
$1 (TABLE 13.1)
13-11
Periods (N) = 3 x 1 = 3
FUTURE VALUE OF AN ORDINARY ANNUITY
Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%.
Rate (R) = 8%/1 = 8%3.2464 (table factor) x $3,000 = $9,739.20
13-12
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.
13-13
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 Yr 1 Interest3,240.00 3,000.00 Beginning Yr 26,240.00
499.20 Yr 2 Interest6,739.20 3,000.00 Beginning Yr 39,739.20
779.14 Yr 3 Interest10,518.34 End of Yr. 3
13-14
CALCULATING FUTURE VALUE OF AN
ANNUITY DUE BY TABLE LOOKUPStep 1. Calculate the number of periods and rate per
period. Add one extra period.
Step 2. Look up in an ordinary annuity table the periods and rate. 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.
13-15
FUTURE VALUE OF AN ANNUITY DUE
Find the value of an investment after 3 years for a $3,000 annuity due at 8%.
Periods (N) = 3 x 1 = 3 + 1 = 4
4.5061 (table factor) x $3,000 = $13,518.30
Rate (R) = 8%/1 = 8%
$10,518.30$13,518.30 -- $3,000 =
13-16
$0.00
$0.50
$1.00
$1.50
$2.00
$2.50
$3.00
$3.50
1 2 3
Number of periods
$.9259
$1.7833
$2.5771
PRESENT VALUE OF AN ANNUITY OF $1 AT 8% (FIGURE 13.2)
13-17
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 the present value of an 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 Present value ofordinary annuity payment payment ordinary annuity table
= x
13-18
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)
PRESENT VALUE OF AN ANNUITY OF $1
(TABLE 13.2)
13-19
PRESENT VALUE OF AN ANNUITY
John Fitch wants to receive a $8,000 annuity in 3 years. Interest on the annuity is 8% semiannually. John will make withdrawals at the end of each year. How much must John invest today to receive a stream of payments for 3 years.
N = 3 x 1 = 3 periods
Manual Calculation20,616.80$ 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 ==>
R = 8%/1 = 8%
2.5771 (table factor) x $8,000 =
$20,616.80
13-20
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 periods
N = 6 x 2 = 12 periods
Step 1.
Step 2.R = 8%/2 = 4%
12.0061 (table factor) x $200 =$2,401.22
Future value of an annuity
Future value of a lump sum
R = 8%/2 = 4%
1.6010 (table factor) x $2,401.22 =
$3,844.35
13-21
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 periods R = 5%/1 = 5%
Step 1. Present value of an annuity Step 2. Present value of a lump sum
R = 5%/1 = 5%
14.0939 x $40,000 = $563,756
N = 8 x 1 = 8 periods
.6768 x $563,756 = $381,550.06
13-22
SINKING FUNDS (FIND PERIODIC PAYMENTS)
Sinking fund = Future x Sinking fund
payment value table factor
Sinking fund –financial arrangement that sets aside regular periodic payments of a particular amount of money
13-23
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
SINKING FUND TABLE BASED ON $1 (Table 12.3)
13-24
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 periods
Check
Future Value of an annuity table
N = 18, R= 10%
* Off due to rounding
R = 10%/1 = 10%
0.0219 x $60,000 = $1,314
$1,314 x 45.5992 = $59,917.35*
